Albedo-Ice Regression method for determining ice water content of polar mesospheric clouds using ultraviolet observations from space

High spatial resolution images of polar mesospheric clouds (PMCs) from a camera array on board the Aeronomy of Ice in the Mesosphere (AIM) satellite have been obtained since 2007. The Cloud Imaging and Particle Size Experiment (CIPS) detects scattered ultraviolet (UV) radiance at a variety of scattering angles, allowing the scattering phase function to be measured for every image pixel. With well-established scattering theory, the mean particle size and ice water content (IWC) are derived. In the nominal mode of operation, approximately seven scattering angles are measured per cloud pixel. However, because of a change in the orbital geometry in 2016, a new mode of operation was implemented such that one scattering angle, or at most two, per pixel are now available. Thus particle size and IWC can no longer be derived from the standard CIPS algorithm. The Albedo-Ice Regression (AIR) method was devised to overcome this obstacle. Using data from both a microphysical model and from CIPS in its normal mode, we show that the AIR method provides sufficiently accurate average IWC so that PMC IWC can be retrieved from CIPS data into the future, even when albedo is not measured at multiple scattering angles. We also show from the model that 265 nm UV scattering is sensitive only to ice particle sizes greater than about 20–25 nm in (effective) radius and that the operational CIPS algorithm has an average error in retrieving IWC of −13± 17 %.


Introduction 19
Polar Mesospheric Clouds (known as noctilucent clouds in the ground-based 20 literature) have been studied for over a century from high-latitude ground 21 observations, but only since the space age have we understood their physical nature, 22 as water-ice particles occurring in the extremely cold summertime mesopause region. 23 Their seasonal and latitudinal variations have now been well documented (DeLand et 24 al., 2006). Interest in these clouds 'at the edge of space' has been stimulated by 25 suggestions that they are sensitive to global change in the mesosphere (Thomas et al., 26 1989). This expectation has been supported recently by a time series analysis of Solar 27 Backscattered Ultraviolet measurements of PMC (Hervig et al., 2016) and by model 28 calculations (Lübken et al., 2018). 29 The Aeronomy of Ice in the Mesosphere satellite (AIM)  was 30 designed to provide a deeper understanding of the basic processes affecting PMC, 31 through remote sensing of both the clouds and their physical environment 32 (temperature, water vapor, and meteor smoke density, among other constituents). 33 One of the two active experiments on board AIM is a camera array, the Cloud Imaging 34 and Particle Size (CIPS) experiment, which provides high spatial resolution images of 35 PMC (McClintock et al., 2009). CIPS measures scattered ultraviolet (UV) sunlight in the 36 nadir in a spectral region centered at 265 nm, where ozone absorption allows the 37 optically-thin ice particles to be visible above the Rayleigh scattering background 38 issuing from the ~50-km region . Because of its 39 wide field of view and 43-second image cadence, CIPS views a cloud element multiple 40 times in its sun-synchronous orbital passage over the polar region, thus providing 41 consecutive measurements of the same location at multiple (typically seven) 42 scattering angles (SA). Together with scattering theory, the brightness of the cloud 43 (albedo) at multiple angles provides constraints needed to estimate the mean ice 44 particle size (Lumpe et al., 2013). From the particle size and albedo measurements, 45 the ice water content is calculated for each cloud element (7.5 x 7.5 km 2 in the most 46 recent CIPS retrieval algorithm). However, over time, the AIM orbit plane has drifted 47 from its nominal noon-midnight orientation to the point where the satellite is 48 currently operating in a terminator orbit. Responding to this altered geometry and the 49 desire to broaden the scope of AIM, new measurement sequences were implemented 50 to provide observations of the entire sunlit hemisphere, rather than just the 51 summertime high-latitude region. Because the total number of images per orbit is 52 fixed by data storage limitations, a new mode (the 'continuous imaging mode') of 53 observations, with a reduced three-minute image cadence, was implemented in 54 February 2016. The present sampling in a single Level 2 pixel contains many fewer 55 scattering angles (often only one). To maintain consistency in the study of inter-56 annual variations of PMC, this necessitates a revised method of retrieving ice water 57 content (IWC) where only a single albedo measurement is available. IWC is a valuable 58 measure of the physical properties of PMC since it largely removes the effects of 59 scattering-angle geometry, is a convenient PMC climate variable when averaged over 60 season, and can be used in comparing with contemporaneous measurements of PMC 61 that use different observational techniques. 62 The Albedo-Ice Regression (AIR) method was developed to fill the need to retrieve 63 PMC IWC with only a single measurement of albedo. Based on the simple notions that 64 both albedo and IWC depend linearly upon the ice-particle column density, multiple 65 linear relationships are established between IWC and cloud directional albedo, 66 depending upon scattering angle. Ice Experiment (SOFIE), which provides IWC and particle sizes. These three sources 72 provide many thousands of albedo-IWC-particle size combinations, from which the 73 AIR regressions are derived. Although the AIR method may be inaccurate for a single 74 retrieval of IWC, averages over many observations result in close agreement as the 75 number of data points increases. The utility of AIR thus depends upon the availability 76 of large data sets that apply to roughly the same atmospheric conditions. For example, 77 we will show CIPS results for July and January averages for ascending and descending 78 portions of the orbit. 79 In this paper we first describe the theoretical framework relating the scattered 80 radiance to mesospheric ice particles. It is desirable to use model data to test the AIR 81 method, without the complications of cloud heterogeneity and viewing geometry. We 82 utilized a first-principles microphysical model that accurately simulates large numbers 83 of cloud properties (number density and particle size distribution). The processes 84 treated by the model include meteor 'smoke' nucleation, growth, and sedimentation, 85 occurring in a saturated environment at density and temperature conditions provided 86 by the main global climate model (Bardeen et al., 2010). Several runs for one-day and 87 multiple-day periods during summer solstice conditions for solar conditions applying 88 to 1995 were analyzed. Cloud radiances (albedos) at 265 nm were calculated for the 89 SA range encountered by the CIPS experiment. We chose a set of cloud simulations to 90 derive a single set of two AIR coefficients through linear regression. The accuracy of 91 the AIR approximation was then tested on the same data, and on other model runs, 92 using averages as a function of SA, and increasing IWC threshold values. Thresholding 93 is necessary to account for the fact that different measurement techniques have 94 different detection sensitivities. This is not a signal/noise issue, rather the ability to 95 discriminate PMC against a background that is usually larger than the PMC signal 96 itself. We show in particular how seasonal means of IWC can be derived from Solar 97 Backscatter Ultraviolet Spectrometer (SBUV) radiance data, without the need to 98 derive particle size. 99 Having tested the technique for model data, we use the same approach with real-life 100 PMC data collected from CIPS in the normal pre-2016 operating mode. This mode 101 provided scattering angles needed to define an ice scattering phase function, from 102 which mean particle size was derived based on assumed properties of the underlying 103 size distribution (Lumpe et al., 2013 We then employed highly-accurate data from SOFIE for ice column density and mean 113 particle size. Since the SOFIE technique uses near-IR solar extinction, it is necessary to 114 derive scattered radiances from the same algorithm used by CIPS. This exercise was 115 performed primarily to test whether the derived AIR results are broadly consistent 116 with those derived from the model and CIPS. 117 After describing the AIR method, we discuss briefly the application of the method to 118 a third contemporaneous experiment, the SBUV satellite experiment, which has in 119 common the same limitations as CIPS in its continuous-imaging mode, namely that 120 measurements of nadir albedo are made at a single scattering angle. This has already 121 resulted in a publication (DeLand and Thomas, 2015) where we provided a time series 122 of PMC IWC from the AIR method extending back to the first SBUV experiment in 123 1979. 124

Theoretical Basis 125
Here we provide a brief overview of the theoretical basis of the IWC retrieval 126 technique, referring to previous publications for more detail (Thomas and McKay, 127 1985;Rusch et al., 2009;, Lumpe et al., 2013. The basic 128 measurement is PMC cloud radiance (Φ, ) where Φ is the scattering angle (angle 129 between the sun and observation vectors) and is the view angle, which is the angle 130 subtended by the nadir and observation direction, measured from the point of 131 scattering. Since the ice layer is optically thin, and secondary scattering is negligible, 132 the albedo is described by first-order scattering. The ratio of scattered (detected) 133 radiance to the incoming solar irradiance * is the albedo * , where 134 (1) Here and are the height and particle radius variables, and zb and zt define the 135 height limits of the ice layer, with the majority of the integrand extending between 83 136 and 85 km. rmin and rmax are particle radii which span the particle size regime 137 responsible for scattering (from ~20 nm to ~150 nm). * is the monochromatic 138 scattering cross-section (cm 2 -sr -1 ) at wavelength l and scattering angle . 139 n(r',z')dr'dz' is the number density of ice particles (cm -2 ) in the ranges r',r'+dr' and 140 z',z'+dz'. For CIPS measurements, each camera has a finite bandpass, centered at 265 141 nm, and is characterized by a function * with an effective width of 10 nm 142 . The albedo derived from this instrument is given by 143 In the model, the ice particles are assumed spherical, but the scattering theory should 144 take account of the non-spherical nature of ice crystals. The best agreement of theory 145 with near-IR mesospheric ice extinction occurs for a randomly rotating oblate-146 spheroid shape, of axial ratio two (Hervig and Gordley, 2010). This shape is assumed 147 in the calculation of * , which is accomplished through a generalization of Mie-Debye 148 scattering theory, the T-matrix method (Mishchenko and Travis, 1998). The radius in 149 the T-matrix approach is defined as the radius of the volume-equivalent sphere. In the 150 model calculations, we will ignore the view angle effect. In the reported CIPS data, 151 the secθ factor is applied to the reported albedos, so that always refers to the nadir 152 albedo ( = 0). 153 The ice water content (IWC) is the integrated mass of ice particles over a vertical 154 column through the ice layer. Its definition is 155 (3) denotes the density of water-ice at low temperature (0.92 g-cm -3 ). Anticipating the 156 results of this study that IWC is linearly related to the column density of ice particles, 157 , we explore the physical basis of this result. As first pointed out 158 by Englert and Stevens (2007), such a relationship exists for certain SA values, for 159 which ~8, in which case it is easily seen that Eq.
(2) is proportional to IWC. 160 However, we find that a linear approximation is valid for a much wider range of 161 scattering angles. To understand this result, we imagine that all particles have the 162 same radius, so that = : ( − : ), where is the Dirac -function. Then Eqs. (1)  163 and (3) 'collapse' to simpler results, 164 Here where is the effective vertical thickness of the ice layer. 165 Eliminating the column density , the ice water content is written 166 (5) denotes the particle volume. Thus in this special case, . A 167 superposition of the effects of all participating particle sizes will exhibit a similar 168 proportionality. When is integrated over all , the contributions from each 169 size are straight lines, each having different intercepts and slopes. 170 well the cloud radiance data can be separated from the bright Rayleigh-scattered 177 background. The CIPS experiment retrieval method relies upon high spatial resolution 178 over a large field of view, and the differing scattering-angle dependence of PMC and 179 the Rayleigh-scattering background (Lumpe et al., 2013). The SBUV retrieval relies 180 upon differing wavelength-dependence of PMC and background, but primarily on the 181 PMC radiance residuals being higher (2 sigma) than fluctuations from a smoothly-182 varying sky background (Thomas et al., 1991;DeLand and Thomas, 2015). The AIM 183 SOFIE method is very different, being a near-IR solar extinction measurement in 184 multiple wavelength bands. SOFIE can detect much weaker clouds than either CIPS or 185 SBUV. Particle radii values as small as 10 nm can be retrieved from the SOFIE data 186 . To compare the various experiments, it is necessary to 'threshold' 187 the data from more sensitive experiments with a cutoff value of IWC. 188 In the next three sections, we present the AIR results from the model, CIPS and SOFIE, 189 using averages over many cloud occurrences. It is not our intention to compare the 190 different thresholded data sets to one another (this task will be relegated to a 191 separate publication), but to illustrate how even measurements made at a single 192 scattering angle (e.g., SBUV) can yield averaged IWC values that are sufficiently 193 accurate to assess variations in daily and seasonal averages. These variations are of 194 crucial value to determining solar cycle and long-term trends in the atmospheric 195 variables (mainly temperature and water vapor) that control ice properties in the cold 196 summertime PMC region. We examine the accuracy of AIR through simulations of 197 scattered radiance from the model, and from CIPS and SOFIE data. Since these data 198 sources yield particle radii, they can provide both the 'actual' and approximate values 199 of IWC from the regression formulas. Hervig and Stevens (2014) used the spectral 200 content of the SBUV data to provide limited information on particle size. Together 201 with the albedos themselves, they used this information to derive seasonally-202 averaged ice water content. They showed that the variation of mean particle size over 203 the 1979-2013 time period was relatively low (standard deviation of ±1 nm). They 204 also found a very small systematic increase with time, as discussed in Sect. 3. 205

1 Model Results 206
Using a microphysical model as a reference source of IWC 'data' is useful, in the 207 following ways: (1) in contrast to the CIPS and SOFIE retrieval algorithms, no artificial 208 assumptions are needed concerning the size distribution of ice particles; (2) radiance 209 and IWC may be calculated accurately, so that effects of cloud inhomogeneity are 210 absent; and (3) limitations due to background removal are absent. In addition, to gain 211 insight into the accuracy of the AIR approach, it is sufficient to work with 212 monochromatic radiance at the central wavelength of the various passbands. The 213 integrations of Eqs. (1) and (3) were approximated by sums over 1-nm increments of 214 radius, and over all sub-layers within the model ice cloud (a typical ice layer is several 215 km thick.). The model height grid is variable, being highest in the saturated region 216 where the smallest layer thickness is 0.26 km (see Bardeen et al., 2010 for more 217 details). We then performed the linear regression for SA values over which CIPS 218 observations are made. 219 clouds. The units of IWC are g-km -2 , or µg-m -2 , which are commonly used in the 221 literature. Each plot is divided into two groups according to the effective radii reff for 222 each cloud. reff is defined in the literature (Hansen and Travis (1974) as 223 (6) Figure 1 clearly illustrates that particle size contributes to the 'scatter' from the linear 224 fits. It also shows the existence of a non-zero intercept of IWC vs albedo. The non-zero 225 intercept was at first surprising since we expected that for an albedo of zero, IWC 226 should also be zero. In fact, we found that the linear relationship breaks down for very 227 small albedo, and the points in the plot narrow down as * ® 0 (not shown). In albedo 228 units of 10 -6 sr -1 (hereafter referred to as 1 G) this departure from linearity occurs for 229 A<1 G and IWC<10 g/km -2 , conditions which fortunately are below the sensitivity 230 threshold of CIPS and SBUV, and therefore unimportant for our purposes. For more 231 sensitive detection techniques, this limitation must be kept in mind. A limitation of 232 the present model (not necessarily all models) is that it does not simulate the largest 233 particles in PMC and the largest values of IWC, as seen in both AIM SOFIE and CIPS 234 experiments. The largest model IWC value is 180 g-km -2 and the largest effective 235 radius is 66 nm, whereas CIPS and SOFIE find particle radii up to 100 nm and IWC up 236 to 300 g-km -2 . This limitation is irrelevant for the AIR CIPS results (to be discussed), 237 but could limit the application of the AIR technique to SBUV data. In Sect. 3 we will 238 return to the issue of the AIR accuracy, as applied to SBUV data. 239 We chose to use averages for the entire model run, which includes different latitudes, 240 longitudes, and UT, but the data can be subset in many different ways. It is certainly 241 preferable in data sets to choose a small time and space interval over which 242 temperature and water vapor are not expected to vary, but this is not necessary for 243 the model. All that we ask of the model is whether the AIR results provide an accurate 244 estimate of < >, taken over the ensemble of model cloud albedos calculated at a 245 variety of scattering angles. 246 As discussed above, we are also interested in the accuracy of AIR in the thresholded 247 data, that is, how AIR represents <IWC> in comparisons of data sets with varying 248 detection sensitivities to PMC. Figure 2 displays the error in the ensemble-average 249 (2488 model clouds) as a function of the IWC threshold and scattering angle. Despite 250 the large data scatter from the linear fit shown in Fig. 1, the averaging removes almost 251 all the influence of the 'random error'. In this case, the overall error is less than 3%. 252 The influence of particle size is of course not a random error, but acts like one in the 253 averaging process. However, the AIR coefficients also depend weakly upon the mean 254 effective radius, defined in Eq. (6) for a single cloud, which varies from one latitude to 255 another and from year to year. The effect of variable on the AIR error is discussed 256 in Section 3. 257

AIR Results from CIPS 258
A detailed description of the Version 4.20 CIPS algorithm, together with an error 259 analysis of individual cloud observations, was presented in Lumpe et al. (2013). Here 260 we describe only what is necessary to understand how IWC is derived from the data.

261
Even though an accurate determination of the scattering-angle dependence of 262 radiance (often called the scattering phase function) is obtained by seven 263 independent measurements, this does not fully define the distribution of particle 264 sizes. Instead, additional constraints need to be introduced to derive the mean 265 particle size. The particles are assumed to be the same oblate-spheroidal shape as 266 defined for the model calculations, and to have a Gaussian size distribution (see eq. 267 11 in Rapp and Thomas (2006). A relationship between the Gaussian width s and the 268 mean particle radius rm is derived from a relationship found in vertically-integrated 269 lidar data (Baumgarten et al., 2010). The net result is that two parameters, the mean 270 particle size and the Gaussian width, are retrieved from a given scattering phase 271 function. However, there is only one independent variable, since the two are related 272 by s(rm). Thus Eq.
(3) simplifies to 273 V denotes the mean ice particle volume evaluated at ? . refers to the retrieved 274 albedo, corrected to view angle θ = 0 and interpolated to scattering angle Φ = 90 o .

275
Note the resemblance of Eq. (7)  Before discussing the AIR results, we first apply the CIPS algorithm to the model data 281 to test how well it works on a set of realistic particle sizes. As mentioned earlier, UV 282 measurements of ice particles are not sensitive to particle radii < 20-25 nm. We 283 applied the CIPS algorithm to 6672 model clouds, using seven scattering-angle points, 284 spanning the range 50 o -150 o (the results are insensitive to the values chosen). We 285 then calculated the % difference between the exact model calculation of IWC and the 286 simulated CIPS retrieved IWC for every model cloud. Figure 3 shows the result as a 287 function of (Φ = 90 A ). The mean difference and standard deviation for two model 288 days is -13±17%. With the caveat that not all ice is retrieved, only a large subset of 289 CIPS IWC data have an acceptable accuracy (an average of 84% of the modelled ice 290 mass is contained in particles with radii exceeding 23 nm). We note that IWC in the 291 model used to derive the AIR approximation refers to all particle sizes. 292 The procedure for deriving AIR coefficients from the CIPS data is as follows: (1) 293 Regression coefficients were derived from data pertaining to 0-40 days from summer 294 solstice (day from solstice, DFS=0 to 40) on every third orbit. This meant that ~200 295 orbits per season were used. The regression analysis was performed on four years of 296 data (2010-2013). The data were binned in 5-degree SA bins and only the best quality 297 pixels with six or more points in the phase function were used; (2) Data from each 298 northern and southern summer season were treated separately. The coefficients and 299 standard deviations of the fit were then interpolated to a finer SA grid from 22° to 300 180° in increments of 1 o ; (3) The coefficients from each hemisphere were averaged, 301 and these coefficients were then used to create an AIR IWC data base, to accompany 302 the normal CIPS products. As previously shown, the AIR data applies to the ice mass 303 of 'UV-visible' clouds, not to their total ice mass. 304 We emphasize that using the AIR data is unnecessary for seasons prior to the northern 305 summer season of 2016 -however the AIR data have great importance since that time 306 because the observing mode was changed, resulting in measured phase functions that 307 contain many fewer (and often only one) scattering angles. As illustrated in Fig. 4, it is 308 trivial to infer both IWC and A(90°) from a single measurement of albedo. This 309 alternative 90-deg albedo value, ALB_AIR, is now included along with IWC AIR in the 310 CIPS Level 2 data files. Fig. 5 shows the AIR results for monthly-averaged IWC (July and 311 January) compared to the same averages of the more accurate results from the 312 operational (OP) retrieval described in Lumpe et al. (2013). The data have been 313 separated into different hemispheres, and into ascending and descending nodes of 314 the sun-synchronous orbit, and apply to the years of the nominal operating mode. The 315 ALB_AIR results are systematically higher than the operationally retrieved 90-deg 316 albedo, whereas there is no consistent bias in the IWC (AIR) value compared to the 317 operational product. However, for both quantities the interannual changes between 318 the AIR and OP results agree very well. This is reflected in the very high correlation 319 coefficients of the two sets of values. A more stringent test of the AIR method comes 320 from daily values of CIPS IWC. Shown in Figs. 6 and 7 are polar projections of IWC (AIR) 321 and the more accurate operational IWC data product. These 'daily daisies' are taken 322 from overlapping orbit strips pertaining to 28 June of two different years. Figure 6  323 shows data from 2012, when CIPS was still in normal mode, and the AIR result shows 324 remarkable agreement with the operational IWC data. By 2016 (see Figure 7) CIPS is 325 in continuous imaging mode and the standard IWC retrieval is limited due to the 326 scarcity of pixels with three or more scattering angles. Here the AIR approach is clearly 327 superior and does a good job of filling in the polar region where CIPS detects high-328 albedo clouds. The differences in patterns are due primarily to variations of particle 329 size, rather than errors in the AIR method. 330 AIR accuracy can also be tested in the study of latitudinal variations. Figure 8  331 compares

Results from SOFIE 337
A third independent data set of IWC and particle size is available from the AIM SOFIE 338 experiment. SOFIE provides very accurate values of IWC, through precise near-IR 339 extinction measurements, independent of particle size. It assumes the same Gaussian 340 distribution of particle sizes as CIPS, so that the reported value of mean particle radius 341 ? is consistently defined. SOFIE data are useful to investigate the extent to which the 342 AIR approximation can be applied to an independent data set. To do so, it is necessary 343 to calculate 265-nm albedo at various SA values, given the values of ? , ice column 344 density from the data base, and the mean cross-section * ( ? , Φ). The latter 345 quantity is averaged over the assumed Gaussian distribution. The equation for the 346 albedo is 347 Given * (Φ, 0) and IWC for each PMC measurement (one occultation per orbit), we 348 can once again perform regressions and find AIR coefficients for the SOFIE data set.

349
The comparison of AIR results from all three data sets is shown in Fig. 9, where the 350 Atmos. Meas. Tech. Discuss., https://doi.org/10.5194/amt-2018-330 Manuscript under review for journal Atmos. Meas. Tech. Discussion started: 9 November 2018 c Author(s) 2018. CC BY 4.0 License. constant term C is the y-intercept and S is the slope in the AIR regression 351 352 (9) Figure 10 displays the results from the three data sets, expressed as contour plots of 353 AIR-derived IWC as functions of SA and Albedo. This comparison shows that the three 354 sets of IWC resemble one another far better than would be anticipated from the AIR 355 coefficients in Fig. 9, where the constant coefficient differs significantly between data 356 sets. Since the result of the regression in yielding IWC is more significant than the 357 coefficients themselves, the comparisons of Fig. 10 are the more appropriate 358 diagnostic. The fact that the IWC derived from AIR is more accurate than would be 359 expected from the differing coefficients is due to the fact that the errors of the 360 constant and slope coefficients are anti-correlated. The agreement between the three 361 results will be even better when taken over a large data set with variable SA and 362 albedo. The comparisons of IWC from different satellite experiments as a function of 363 year and hemisphere will be the subject of a separate publication. 364 Figure 11 shows that the regressions with AIM SOFIE data obey a linear relationship 365 between IWC and albedo for IWC <220 g-km -2 , but for SA values <90 o , AIR 366 overestimates IWC by up to 15%, depending upon the SA. For SA=100 o the regressions 367 are still linear up to 300 g-km -2 , values above which are seldom encountered in the 368 data. 369

SBUV data 370
The AIR coefficients from the model have been used by DeLand and Thomas (2015) to derive 371 mean IWC from SBUV data, which spans the largest time interval of any satellite data set 372 (1979-present). The 273 nm wavelength used in the SBUV Version 3 analysis is sufficiently 373 close to the effective wavelength of the broader passband of the CIPS cameras (Benze et al., 374 2009) that the same coefficients may be applied to both data sets. The accuracy of the average 375 IWC results was estimated by removing half the data from an entire season and comparing 376 the two results. For a highly-populated region (more than 1000 clouds per season at latitudes 377 higher than 70°), the changes in IWC were ±3 − 5 g-km -2 . For a less populated region (50°− 378 64° latitude) where there were many fewer clouds (<50), the changes were larger, ±5 − 10 g-379 km -2 . Even the larger errors are sufficiently small for intercomparison of SBUV and 380 contemporaneous PMC measurements. Figure 12 shows a comparison of SBUV IWC, using 381 the model AIR coefficients, to the results of a more accurate determination of IWC derived 382 from particle size determinations using SBUV spectral information (Hervig and Stevens, 2014).

383
The comparison is for data residuals from July averages over the time series 1979-2017. Given 384 the different assumptions underlying the two data sets, the agreement is very good (with an 385 rms difference of 3% for the residuals, and 5% for the actual values of <IWC>).

Effects of Mean Ice Particle Size 387
The AIR approximation is based on the notion that particle size effects can be ignored. 388 In fact, the particle size (or more accurately, 8 ) is a principal 'driver' of < > 389 itself, so it is not obvious that particle size effects may be neglected. Column density 390 also drives IWC, and the dependence of albedo on density adequately captures this 391 part of the variability (albedo is strictly linear in column density). The AIR slope term 392 is proportional to K L M N (K,O) averaged over a distribution of particle sizes, r. Since 393 * ( , Φ)~8 PQ (where the exponent depends upon Φ) then averaging over many 394 values of r results in a slope term that, in the limit of large number, depends 395 predominantly on Φ. This is an example of "regression to the mean", and illustrates 396 how the approximation is designed to work for large numbers of clouds. In a fictitious 397 case where the mean cloud particle size is larger in one year than another, but the 398 cloud column number remains the same, the mean albedo would increase according 399 to Eq. (8), resulting in an increase of <IWC>. We might expect that the slope term 400 would be different in the two cases. Our study with three different data sets shows 401 that the regression slope itself remains almost the same among the three data sets, 402 despite their differing in mean particle size. 403 In fact, Hervig and Stevens (2014) found from SBUV spectral data a small long-term 404 trend in <IWC> and in addition a trend in the mean particle size (+0.23 ± 0.16 405 nm/decade). This contributed an additional 20% to the overall long-term trend in 406 <IWC>. The ignored dependence on mean particle size using the AIR method thus 407 adds a systematic uncertainty in derived <IWC> trends, which can be as large as 20%, 408 according to Hervig and Stevens (2014). This error undoubtedly varies inversely with 409 the number of clouds in the averaging process. For example, the number of CIPS 410 observations per PMC season greatly exceeds that of SBUV, so that the error in <IWC> 411 should be correspondingly smaller. 412

Conclusions 413
We have described the theoretical basis and accuracy for an approximation for retrieving the suggests that CIPS is capable of measuring 84% of the total ice content of PMC (for particle 421 sizes exceeding ~23 nm). The accuracy for measuring the total (over all particle sizes) IWC is -

472
(<(IWC>-long-term mean) derived by two independent methods from SBUV 273 nm albedo 473 data. Black curve: <IWC> derived from the AIR approximation. Blue curve: <IWC> derived from 474 the same SBUV albedo data, but including mean particle size variations (see text). A three-

475
year smoothing has also been applied.

477
Data Availability

479
The CIPS operational PMC data, along with the AIR data, can be found at