We first examine ambient OC and EC for the purpose of identifying distribution features that can serve as the reference basis for parameterizing the numerical experiments. The 1-year hourly EC and OC measurement data from three sites in the PRD (one suburban site in Guangzhou, a general urban site and a roadside site in Hong Kong, with more than 7000 data at each site), are plotted in Fig. S1 in the Supplement document for the whole year data sets and Figs. S2–S4 for the seasonal subsets using the Nancun site as the example. A brief account of the field ECOC analyzers and their field operation is provided in the Supplement. A detailed description of the measurement results and data interpretation for the sites will be given in a separate paper. The distributions of measured OC, EC and OC / EC are fitted by both normal and log-normal distribution curves and then examined by the Kolmogorov–Smirnov (K–S) test. The K–S statistic, D, indicates that log-normal fits all three distributions better than the normal distribution (D values are shown in Figs. S1–S4). Therefore, log-normal distributions are adopted to define the OC, EC and OC / EC distributions during data generation in our numerical experiments. Statistics of these ambient OC and EC, along with a few other measurements reported in the literature, are summarized in Table 2 and are considered as the reference for data generation to better represent the real situation.

The probability density function (PDF) for the log-normal distribution of variable x is fx;μ,σ=1xσ2π×e-(ln⁡x-μ)22σ2. The two parameters, μ and σ, of the log-normal PDF are related to the average and standard deviation of x through the following equations: μ=ln⁡avg-0.5×ln⁡1+SD2avg2σ=ln⁡1+SD2avg2. First, realistic average and standard deviation values of EC, (OC / EC)pri, and OC (e.g. Figs. S1–S5) are adopted to calculate μ and σ. Then the pseudorandom number generator uses μ and σ to synthesize EC and OC data sets.

The Mersenne twister (MT) (Matsumoto and Nishimura, 1998), a pseudorandom number generator, is used in data generation. MT is provided as a function in Igor Pro. The system clock is utilized as the initial condition for generation of pseudorandom numbers. The data generated by MT have a very long period of 219 937–1, permitting large data size and ensuring that pseudorandom numbers are statistically independent between each data generation. The latter feature ensures the independent relationship between EC and non-combustion related SOC data. The case with combustion-related SOC is briefly discussed in Sect. 3. MT also allows assigning a log-normal distribution during pseudorandom number generation to constrain the data. For the verification of the log-normality of MT generated data, a series of K–S tests on the generated data for 5000 runs are conducted. As shown in Fig. S6, 94.4 % of runs pass the K–S test. Hence the performance of MT can satisfy the log-normal distributed data generation requirement in this study. In a previous study, Chu (2005) used a variant of sine functions to simulate POC and EC, which limited the data size to 120, and the frequency distributions of POC and EC exhibited multiple peaks, a characteristic that is not realistic for ambient measurements. The key information utilized in the EC tracer method is the correlation between EC and POC as well as the irrelevance between EC and SOC. The time series information is not needed in EC tracer method, making pseudorandom number generator a good fit for the evaluation purpose.

The procedure of data generation for the single emission source scenario is illustrated in Fig. 2 and implemented by scripts written in Igor Pro. EC is first generated with the following parameters specified: sample size (n), average and relative standard deviation (RSD%) of the whole data set (see Supplement). The EC data set statistically follows a log-normal distribution, while the sequence of each data point is randomly assigned. POC is then calculated by multiplying EC by (OC / EC)pri (Eq. 1). For simplicity, (OC / EC)pri is set to be a single value, while an analysis incorporating randomly generated log-normally distributed (OC / EC)pri values can be found in the Supplement, and a brief summary is given in Sect. 2.2. SOC data are independently generated in a similar way to that for EC. The sum of POC and SOC then yields the synthesized OC. OC and EC data generated in this way are used to calculate SOC by different implementations of the EC tracer method. The bias of SOC estimation can then be evaluated by comparing the calculated SOC with the “true” SOC values. Data generation for the scenarios with two primary emission sources is similar to the single source scenario and the steps are illustrated in Fig. S7.

Three scenarios are considered. Scenario 1 (S1) considers one single primary emission source. Scenario 2 (S2) considers two correlated primary emission sources, i.e., two sets of EC, POC, and each source has a single but different (OC / EC)pri value. An example of S2 is combined vehicular emissions from diesel-fuel and gasoline-fuel vehicles. These two sources of vehicular emissions have different (OC / EC)pri, but often share a similar temporal variation pattern, making them well correlated. Scenario 3 (S3) considers two independent primary emission sources and simulates an ambient environment influenced by two independent primary emission sources, e.g. local vehicular emissions (lower (OC / EC)pri) and regional biomass burning (higher (OC / EC)pri).

In the following numerical experiments, three (OC / EC)pri estimation methods are examined and compared, including MRS, OC / EC10% and OC / ECmin⁡. As a single point, OC / ECmin⁡, in ambient samples may be subjected to large random uncertainties, thus data with the lowest 1 % OC / EC are adopted instead to derive the OC / ECmin⁡.

In the preceding numerical analysis, the simulated EC and OC are not assigned any measurement uncertainty; however, in reality, every EC and OC measurement is associated with a certain degree of measurement uncertainty. We next examine the influence of OC and EC measurement uncertainty on SOC estimation accuracy by different EC tracer methods. Two uncertainty types are tested, i.e., constant relative uncertainty (Case A); constant absolute uncertainty (Case B). This section mainly focuses on sensitivity tests assuming different degrees of Case A uncertainties. Results assuming Case B uncertainties are discussed in the next section. The uncertainties are assumed to follow a uniform distribution and generated separately by MT. It is also assumed that the uncertainty (εEC or εOC) is proportional to the concentration of EC and OC through the multiplier γunc (i.e., relative measurement uncertainty).

-γuncEC≤εEC≤γuncEC-γuncOC≤εOC≤γuncOC In order to compare the estimated SOC with simulated SOC with εSOC, the measurement uncertainties of POC and SOC are then back-calculated following the uncertainty propagation formula and assuming the same relative measurement uncertainty for POC and SOC (Harris, 2010) γunc′=γuncOC2POC2+SOC2-γunc′POC≤εPOC≤γunc′POC-γunc′SOC≤εSOC≤γunc′SOC. The simulated EC, POC and SOC with measurement uncertainties (abbreviated as ECsimulated, POCsimulated and SOCsimulated respectively) are determined as

ECsimulated=ECtrue+εECPOCsimulated=POCtrue+εPOCSOCsimulated=SOCtrue+εSOC. Sensitivity tests of SOC estimation as a function of relative measurement uncertainty (γunc) and fSOC is performed as shown in Fig. 7 by comparing the estimated SOC with SOCsimulated. Fixed input parameters include n=8000; EC = 2±1 µgC m-3; (OC / EC)pri=0.5. Studies by Chu (2005) and Saylor et al. (2006) suggest that ratio of average POC to average EC (ROA, see Supplement for details) is the best estimator of the expected primary OC / EC ratio because it is mathematically equivalent to the true regression slope when the data contain no intercept. ROA is confirmed as the best representation of (OC / EC)pri for SOC estimation, which shows no bias towards γunc or fSOC change. MRS overestimates SOC and the positive bias increases with γunc while decreasing with fSOC (Fig. 7). The SOC estimates by OC / ECmin⁡ and OC / EC10% exhibit larger bias than those by MRS. For example, as shown in Fig. 7a, when fSOC=20 % and γunc=10 %, the bias of SOC by MRS, OC / EC10% and OC / ECmin⁡ is 8, -28 and 36 %, respectively. With increasing fSOC, the bias of SOC by OC / ECmin⁡ decreases while the bias of SOC by OC / EC10% increases when γunc=10–20 %. MRS always demonstrates the best performance in SOC determination amongst the three (OC / EC)pri estimation methods. When γunc could be controlled within 20 %, the SOC bias by MRS does not exceed 23 % when fSOC=20 % (Fig. 7a). If the fSOC ratio falls in the range of 60–80 % and γunc is < 20 %, the OC / ECmin⁡ has a similar performance as MRS, but SOC by OC / EC10% still shows a large bias (∼ 41 %) (Fig. 7c and d).

Sensitivity studies of SOC estimation as a function of γunc and (OC / EC)pri are performed and the results are shown in Fig. S8. In all the three (OC / EC)pri representations, SOC estimates are sensitive to γunc but insensitive to the magnitude of (OC / EC)pri. In the single primary source scenario (S1), it is proved that the performance of MRS regarding SOC estimation is mainly affected by γunc and to a lesser degree by fSOC. Other variables such as (OC / EC)pri and EC concentration do not affect the accuracy of SOC estimation.

MRS relies on correlations of input variables and it is expected that MRS performance is sensitive to the sample size of input data set. This section examines the sensitivity on sample size by the three (OC / EC)pri representations and aims to provide suggestions for an appropriate sample size when applying MRS on ambient OCEC data. Sample sizes ranging from 20 to 8000 are tested and for each sample size 500 repeat runs are conducted to obtain statistically significant results. Both Case A (i.e., a constant relative uncertainty of 10 %) and Case B (i.e., a constant absolute uncertainty of ±0.2 µgC m-3 for both OC and EC) are considered. The measurement uncertainties in case B are generated separately by MT following a uniform distribution within the range of ±0.2 µgC m-3. The measurement uncertainties of POC and SOC are then back-calculated following the uncertainty propagation formula (Harris, 2010) and assuming the ratio of εPOC / εSOC is the same as POC / SOC ratio (controlled by fSOC).

The mean SOC bias by MRS is very small (< 3 %) for all sample sizes while the standard deviation of SOC bias decreases with sample size (Fig. 8). The standard deviation of SOC bias is ∼ ±30 % at the lowest test sample size (n=20), and decreases to less than ±15 % at n=60 (the sample size of 1-year sampling from an every-6-day sampling program) and to less than ±10 % at n=200. Similar patterns are observed between Case A (Fig. 8a) and Case B (Fig. 8b) for MRS and OC / EC10%. For OC / ECmin⁡, a larger bias is observed in Case B than Case A for all sample sizes, as SOC bias by OC / ECmin⁡ is more sensitive to measurement uncertainty in the range of 0–10 % as shown in Fig. 7b. The standard deviation of SOC bias by OC / ECmin⁡ and OC / EC10% both decrease with sample size as shown in Fig. 8. The mean SOC bias of OC / ECmin⁡ decrease with increased sample size while OC / EC10% is insensitive to sample size. The sample size dependency of all three (OC / EC)pri representations is not sensitive to fSOC as shown in Fig. S16. Other scenarios considering (OC / EC)pri with a distribution and different fSOC are discussed in the Supplement.

Besides hourly measurements of OC and EC by online aerosol carbon analyzers, the MRS method could also be applied to offline measurements of OC and EC based on filters collected over longer durations (i.e., 24 h), which are more readily available around the world. To explore the impact of sampling duration (e.g., hourly vs. daily), we here use 1-year hourly data at the suburban site of Guangzhou to average them into longer intervals of 2–24 h. The 24 h averaged samples yield a (OC / EC)pri of 2.53, 12 % higher than the (OC / EC)pri derived from hourly data (2.26). This comes as a result of that OC / EC distributions are narrowed when the averaging interval lengthens (Fig. 9), leading to elevation of the MRS-derived (OC / EC)pri. As many PM2.5 speciation networks adopt a sampling schedule of one 24 h sample every 6 days, we further extract the every-6-day samples to do the MRS calculation. The 1-year data yield six subsets of daily samples, corresponding to six possible schedules of sampling days with the every-6-day sampling frequency. The MRS calculation produces the OC / ECpri in the range of 2.37–2.75 (5–22 % higher than the OC / ECpri from the hourly data). This example illustrates that if 24 h sample ECOC data are used, SOC would be biased slightly lower in comparison with those derived from the hourly data.