Theory
3.3 Data Evaluation
3.3.2 Size Distribution Retrieval
Cloud-free time intervals with homogeneous sampling conditions, i.e. constant alti-tude flight intervals with homogeneous particle concentrations, are used to derive the aerosol particle NSD from the combination of CPC and OPC measurements. As (most) prevalent methods for the derivation of NSDs from OPC measurements are inadequate, especially concerning a proper uncertainty analysis (Walser et al., 2017), within the scope of this study a new NSD retrieval method has been developed. As explained in detail in the following, this NSD retrieval method is self-consistent with the OPC calibration approach presented in Sect. 3.2.2.2, addresses the actual inverse problem underlying OPC measurements and allows for a thorough propagation of all uncertain-ties in the underlying parameters.
As illustrated in Fig. 3.3.1 and previously proposed by Fiebig et al. (2005), the NSD retrieval is based on a Monte Carlo method. It starts with drawing random samples from all initial parameters being subject to systematic uncertainty. A complete list of these parameters with associated PDF assumptions is given in Tab. 3.3.1. In Fig. 3.3.1 the parameter PDFs and the Monte Carlo sampling from these PDFs are visualized by dashed boxes and arrows, respectively. For the CPC, the parameters afflicted with systematic uncertainty include a, D1 and D2 determining the size-dependent count-ing efficiency via Eq. (3.2.3), and the potential concentration deviation (∆n/n)sys, e.g.
caused by an eventual small deviation in the sample flow from the value used to convert counts to concentration. Using the Monte Carlo picks from the corresponding PDFs and the measured interval mean concentration ¯n, the CPC counting efficiencyηCPC(D) and a shifted mean concentration nCPC = ¯n·1 + (∆n/n)sys are calculated. For the OPCs, the uncertainty in the instrument response is given by the model parameter solution ensemble resulting from the evaluation of the calibration measurements, as described in Sect. 3.2.2.2. In contrast to the CPC, additional uncertainty is intro-duced by the optical properties of the aerosol particles, here taken into account by the bulk refractive index m and an additional OPC signal broadeningbas induced by particle asphericity. From the OPC response model parameter pick (b, s, c, σε) and the additional bas a combined response tuple is derived as (btotal, s, c, σε), with the total signal broadening btotal = qb2+b2as being the standard deviation of the convo-lution of the two individual (Gaussian) broadening effects. The bulk refractive index pick m is used to calculate the size-dependent theoretical (midway) scattering cross section ˜Cscat,m(D) for the specific OPC detection geometry via Mie theory. In a sub-sequent step, a random shift from this theoretical function is performed according to its potential (time-dependent) systematic deviation, e.g. induced by light source in-tensity fluctuations. This relative shift ε is drawn from the Gaussian PDF N(1, σε2) and yields the shifted ˜Cscat,m,ε(D) = ε·C˜scat,m(D). From ˜Cscat,m,ε(D), the instru-ment parameter tuple (btotal, s, c) and the known OPC bin threshold values Ui the OPC kernel functions Υi(D) are calculated via Eq. (3.2.10). Under the justified as-sumption that the OPC sampling efficiencies do not significantly deviate from unity,
Figure 3.3.1: Flowchart for the NSD retrieval. Each Monte Carlo iteration starts with random samples (picks) from the initial parameter PDFs (1), that are subsequently used to calculate particle concentrations (2) and corresponding counting efficiency functions (3). The results are input to the NSD inversion (4) that is performed by means of a MCMC method. The NSD inversion results for each Monte Carlo iteration are finally merged into a collective NSD solution ensemble. The multiplicand f in step (2) is defined by f = 1 + (∆n/n)sys.
parameter(s) description random Monte Carlo sampling from...
CPC1
a,D1 andD2 CPC counting efficiency
function parameters (uncorrelated) Gaussian PDFs with 3%, 7%
and 2% relative standard deviation fora, D1 and D2, respectively (cf. Tab. 3.2.1) (∆n/n)sys systematic concentration
deviation Gaussian PDF centered at zero with 5%
standard deviationN(0,0.052) UHSAS
(b, s, c, σε) OPC response model
parameters parameter solution ensemble resulting from the OPC calibration evaluation ε systematic scattering cross
section deviation Gaussian PDFN(1, σε2) defined by the previousσε pick
bas additional signal
broadening due to particle asphericity
Gaussian PDFN(0.06,0.032) for the SALa
m (bulk) particle refractive
index uniform PDFU([1.5,1.6] +i[0,0.005]) for the SAL,
uniform PDFU([1.4,1.5] +i0) for the MBL
(∆n/n)sys systematic concentration
deviation Gaussian PDFN(0,0.052) SkyOPC
(b, s, c, samp, camp, σε) OPC response model
parameters parameter solution ensemble resulting from the OPC calibration evaluation ε systematic scattering cross
section deviation Gaussian PDFN(1, σε2) defined by the previousσε pick
bas additional signal
broadening due to particle asphericity
Gaussian PDFN(0.09,0.042) for the SALa
m (bulk) particle refractive
index uniform PDFU([1.5,1.6] +i[0,0.005]) for the SAL,
uniform PDFU([1.4,1.5] +i0) for the MBL
(∆n/n)sys systematic concentration
deviation Gaussian PDFN(0,0.052)
aDue to hydration particles in the MBL are assumed to be (close to) spherical shape, thusbas= 0.
Table 3.3.1: Parameters that are sampled via a Monte Carlo method in the course of the NSD retrieval (see step (1) in Fig. 3.3.1) together with their PDFs, that define the corresponding systematic uncertainties.
Figure 3.3.2: Exemplary set of counting efficiency functions ηk(D) calculated for each NSD retrieval iteration (see step (3) in Fig. 3.3.1). The example represents the entirety of counting efficiency functions for the CPC1, the SkyOPC and UHSAS bins using the best estimates for the instrument parameters, assuming spherical particles (bas = 0) with a refractive index of m = 1.54 +i0 and no relative deviation from the resulting scattering cross sections (ε= 0).
the kernel functions directly correspond to the bin counting efficiency functions, i.e.
ηi,OPC(D) = Υi(D). Like for the CPC, the measured mean OPC bin concentrations are further (collectively) shifted according to the estimated range of potential system-atic deviation to obtain the ni,OPC. All resulting CPC and OPC concentrations and counting efficiency functions are then collected to yield the set of nk ={nCPC, ni,OPC} and ηk(D) ={ηCPC(D), ηi,OPC(D)}for the Monte Carlo iteration. An exemplary set of ηk(D) is shown in Fig. 3.3.2. The parameter set is complemented by the relative (statistical) concentration uncertainties SE(¯nk)/n¯k calculated by the standard errors of the means SE(¯nk), i.e. the interval standard deviations of the measured (bin) con-centrations divided by the square root of the number of observations (i.e. time steps).
These uncertainties are caused by counting variability (Poisson statistics) and any re-maining ambient concentration variability15. The counting variability and, therewith, the relative measurement uncertainty decreases with increasing total number of parti-cle counts. This means that SE(¯nk)/n¯k decreases with increasing ¯nk and increasing sampling duration.
The centerpiece of the retrieval is the actual NSD inversion that is realized by means of a nested MCMC process. As for the evaluation of OPC calibration measurements (cf. Sect. 3.2.2.2), the task of this inversion process is to harmonize the measured nk
15Further potential causes for measured concentration variability are sample flow or pressure fluc-tuations in the instruments (around a temporal average), that are unconsidered in the calculation of the (STP) concentrations.
parameter(s) description constraint(s) nmode,0,nmode,1andnmode,2 integral particle number
concentrations of NSD lognormal modes
positive
CMD0,CMD1 andCMD2 count median diameters of
NSD lognormal modes mode position confinement:
0.01≤CMDl≤1µm forl∈ {0,1,2}; ascending mode order and mode separation:
(CMDl+1−CMDl)/CMDl≥0.5 for l∈ {0,1,2}
GSD0,GSD1 andGSD2 geometric standard deviations of NSD
lognormal modes
no (quasi-)monomodal or excessively broad modes:
1.25≤GSDl≤3 forl∈ {0,1,2} dn/dlog10D NSD, i.e. superposition of
NSD modes drop-off to small particle sizes, i.e. no significant occurrence of particles with diameters smaller than the CPC cutoff:
dn/dlog10D≤1 forD≤5 nm
Table 3.3.2: Prior probability assumptions for the parameters defining the multimodal lognormal NSDs.
with their modeled counterparts ˆnk given by Eq. (3.2.8) as ˆ
nk = ˆ+∞
−∞
ηk(D) dn
dlog10Ddlog10D (3.3.1) with the (logarithmic) NSD dn/dlog10D. In contrast to the OPC calibration, how-ever, the inversion process in this case aims to find the solutions for dn/dlog10D. In this study, the NSD is parametrized by three lognormal modes. One reason for this is that representing the NSD as a superposition of lognormal modes is common practice, particularly in aerosol models, another reason is that in doing so the inverse problem reduces to the determination of the nine lognormal mode parameters, i.e. the NSD mode integral number concentrations nmode,l, count median diameters CMDl and ge-ometric standard deviations GSDl. Accordingly, Eq. (3.3.1) can be rewritten as (cf.
Sect. 2.3) ˆ
nk = ˆ∞
0
ηk(D)· 1 Dln 10
2
X
l=0
nmode,l
√2πlog10GSDlexp
−1 2
log10D−log10CMDl log10GSDl
!2
dD (3.3.2) As the measured concentrationsnkfollow a Gaussian PDF, the probabilityP ({nˆk} | {nk}) for the model values to occur given the measurements is symmetric in ˆnk and nk, so that the likelihood function P({nk} | {nˆk}) can be expressed as
P ({nk} | {nˆk}) =Y
k
√ 1
2πσk exp −(nk−nˆk)2 2σk2
!
with the Gaussian standard deviationsσkof the modeled ˆnkinferred from the measured relative concentration uncertainties SE(¯nk)/n¯k as
σk = ˆnkSE(¯nk)
¯ nk
For the same arguments as in Sect. 3.2.2.2, the prior probabilities for the NSD mode parameters P (nmode,l,CMDl,GSDl) are assumed flat, i.e. uniform within physically reasonable limits. The prior probability constraints for the NSD (parameters) are listed in Tab. 3.3.2. As before, the MCMC software tool emcee (Foreman-Mackey et al., 2013) is utilized for the maximization of the (posterior) probability, or in other words for the determination of the probable range of NSD solutions under the given uncertainties. The number of Markov chain (burn-in) iterations is adjusted to guar-antee representative solution ensembles for each maximization process. Repeating the whole retrieval loop many times propagates all initial parameter uncertainties and fi-nally results in a collective (fully correlated) solution ensemble for the nine NSD mode parameters (cf. Fig. A.3.1). This solution ensemble can be seen as a collection of random samples from the final NSD (parameter) PDFs including all systematic and statistical uncertainties.
The initial PDFs for the aerosol optical propertiesmandbas, specified in Tab. 3.3.1, are based on particle composition measurements, certain assumptions and literature values. For the MBL, Kristensen et al. (2016) show that the majority of particles in the size range D ≤ 500 nm are completely volatile, with the measurements indicat-ing a dominance of soluble inorganic sulfates (ammonium sulfate) and a non-negligible contribution from organics, likely biogenic secondary organics (K. Kandler and T. Kris-tensen, personal communication). Contributions from non-volatile particle components such as sea salt, soot or mineral dust are marginal and predominantly appear as small fractions in internal mixtures with the volatile material. For larger particles, sea salt is assumed to play an important role, while the contribution of mineral dust is note-worthy but subordinate16. Literature values for the refractive index of dry ammonium sulfate (at the used OPC wavelengths) range from about 1.50 to 1.53 (Toon et al., 1976;
Tang, 1996; Michel Flores et al., 2012). A similar refractive index is reported for dry sea salt, that is predominantly made up of sodium chloride (e.g. Shettle and Fenn, 1979).
While the refractive index for biogenic secondary organics is subject to some degree of uncertainty, Kim and Paulson (2013) give a most representative value of 1.44. As all these particle types are hygroscopic (to a certain extent), it must be assumed that a considerable fraction of the particles in the humid well-mixed MBL have met relative humidities larger than their DRHs and that they approach the sampling inlets at am-bient relative humidities larger than their efflorescence relative humidities (ERH). This means that these particles are most likely hydrated. Knowing the temperature in the OPC measurement chamber, the relative humidity of the aerosol sample at the point of measurement can be calculated from the ambient temperature and relative humidity.
The resulting range of relative humidities in the OPCs is converted into estimates for the associated particle growth factorsgf =D/Ddry using hygroscopic growth functions reported in the literature (Tang and Munkelwitz, 1994; Gao et al., 2006; Varutbangkul et al., 2006; Cheng et al., 2015). The estimated range for the (bulk) refractive index then follows from these growth factor estimates (gf ≤1.2), the (dominant) dry particle refractive indices and the refractive index of water (e.g. Daimon and Masumura, 2007) using a volume mixing rule. Fierz-Schmidhauser et al. (2010) and Michel Flores et al.
(2012) show that the utilization of a volume mixing rule to calculate the refractive
in-16According to Groß et al. (2016), despite their large size the contribution of mineral dust particles to the total aerosol particle volume in the MBL is<40%.
dex of hydrated particles is often a satisfactory assumption for soluble materials. The imaginary part of the refractive index is assumed to be negligible for particles in the MBL. Because the (majority of) particles are assumed to be slightly hydrated at the point of measurement, they are assumed spherical and, hence,bas is neglected, too.
In the SAL, particle composition measurements, that will be discussed in detail in Sect. 4.1.3 and 4.2.3, identify ammonium sulfate and mineral dust as the major particle components. Since, in contrast to the MBL, ambient relative humidities in the SAL are low and the ambient temperatures fall significantly below the temperatures in the OPC chambers, the relative humidities at the point of measurement certainly are below the ERH for all components, i.e. the particles are measured in dry state. The reported range for the refractive index of African mineral dust covers 1.5 ≤ m ≤ 1.6 with an imaginary part of ≤ 0.005 (e.g. Petzold et al., 2009; Schladitz et al., 2009;
McConnell et al., 2010; Kandler et al., 2011; Formenti et al., 2011a; Ryder et al., 2013b; Denjean et al., 2016a), therewith comprising the refractive index range for the sulfate (see above). OPC signal broadening owing to varying orientations of aspherical particles is estimated from DDA calculations (performed by J. Gasteiger, LMU). These calculations involve a range of particles (cf. Fig. 2.2.4) with aspect ratios≤1.6, being a realistic upper bound for the particle asphericity in the present size range (Kandler et al., 2011). Although bas can principally be size-dependent, for simplicity only one average bas is utilized for each OPC. Anyhow, bas only has a minor effect on the total signal broadening btotal, as the instrument-induced broadening b is significantly stronger.
Despite the fact that the presented NSD retrieval method is one of the most compre-hensive methods to date, some remaining imperfections should not be left unmentioned.
For simplicity and because no exact information on the optical particle properties is available, a potential size-dependence of mis not considered, but the (bulk) refractive index picked for each Monte Carlo iteration determines ˜Cscat,m(D) for the entire size range of each OPC. Moreover, the same m is used for the SkyOPC and the UHSAS, although the two OPCs operate at different wavelengths. Further neglected is the additional signal broadening, that could be induced by external mixtures of particles having the same size but different refractive indices. However, as both in the MBL and the SAL the dominant particle components are assumed to have similar refractive indices (with overlapping uncertainty ranges), the potential size-dependence of m is expected to be only minor, thus having insignificant impact on the NSD solution. For the same reason, additional signal broadening by external particle mixtures is assumed to not cause significant systematic deviations. Last, differences inm owing to the dif-ferent OPC wavelengths are small compared to the overall refractive index uncertainty.
Another potential source of systematic deviation could be the NSD parametrization itself. Although the multimodal lognormal parametrization is a commonly accepted form to express NSDs, the true NSDs might partly deviate from a multimodal lognor-mal shape. The NSD retrieval results must, hence, be viewed in the context of this approximation.
Figure 3.3.3: Flowchart visualizing the derivation of size-dependent particle volatil-ity. Each Monte Carlo iteration starts with random samples (picks) from the initial parameter PDFs (1), that are subsequently used to calculate the refractory particle concentrations for each SkyOPCTDbin (2), the corresponding counting efficiency func-tions (3) and the (modeled) total particle concentrafunc-tions (4). The results allow to determine the volatile particle fractions vfj and mean particle diameters ¯Dj for each bin (5). The set of these discrete pairs obtained for each iteration is finally combined to a collective solution ensemble for the size-dependent volatility.