the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Improved counting statistics of an ultrafine differential mobility particle size spectrometer system

### Dominik Stolzenburg

### Tiia Laurila

### Pasi Aalto

### Joonas Vanhanen

### Tuukka Petäjä

### Juha Kangasluoma

Differential mobility particle size spectrometers (DMPSs) are widely used to measure the aerosol number size distribution. Especially during new particle formation (NPF), the dynamics of the ultrafine size distribution determine the significance of the newly formed particles within the atmospheric system. A precision quantification of the size distribution and derived quantities such as new particle formation and growth rates is therefore essential. However, size-distribution measurements in the sub-10 nm range suffer from high particle losses and are often derived from only a few counts in the DMPS system, making them subject to very high counting uncertainties. Here we show that a CPC (modified Airmodus A20) with a significantly higher aerosol optics flow rate compared to conventional ultrafine CPCs can greatly enhance the counting statistics in that size range. Using Monte Carlo uncertainty estimates, we show that the uncertainties of the derived formation and growth rates can be reduced from 10 %–20 % down to 1 % by deployment of the high statistics CPC on a strong NPF event day. For weaker events and hence lower number concentrations, the counting statistics can result in a complete breakdown of the growth rate estimate with relative uncertainties as high as 40 %, while the improved DMPS still provides reasonable results at 10 % relative accuracy. In addition, we show that other sources of uncertainty are present in CPC measurements, which might become more important when the uncertainty from the counting statistics is less dominant. Altogether, our study shows that the analysis of NPF events could be greatly improved by the availability of higher counting statistics in the used aerosol detector of DMPS systems.

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Differential/scanning mobility particle size spectrometer (DMPS or SMPS) systems can be used to measure the number size distribution of ambient aerosol particles ranging in size from sub-10 nm to hundreds of nanometres (Aalto et al., 2001; Wang and Flagan, 1990). The instruments typically consist of an impactor, a charger, a DMA (differential mobility analyser), and a CPC (condensation particle counter). The impactor is used to limit the maximum particle size to enable multiple charging corrections in the inversion. The charger then brings the particles to a known charge distribution (typically steady-state bipolar charging equilibrium as described by, for example, Wiedensohler, 1988), and the charged particles are size-selected in a DMA based on their electrical mobility. Finally, the number concentration is counted by condensational growth and subsequent optical detection with a CPC. The number size distribution is then determined by stepping/scanning different voltages at the DMA and the application of an inversion process if the maximum particle size, the charging probability, all the losses, and the detection efficiency are known.

As size predominantly determines the dynamics of ultrafine aerosol particles, measurements of the particle number size distribution are essential for understanding the role of aerosols in the atmospheric system. One process in which the smallest ultrafine (< 100 nm) particles are the most important is so-called atmospheric new particle formation (NPF). During NPF, small molecular clusters form from gaseous precursors and subsequently grow to larger sizes (Kulmala et al., 2013), where they can contribute to the budget of cloud condensation nuclei and impact the Earth's radiative balance (e.g. Gordon et al., 2017). To obtain an in-depth understanding of the dynamics of NPF, it is essential to measure the number size distribution down to even sub-10 nm aerosol particles accurately and reliably (Dada et al., 2020; Kulmala et al., 2012). However, there are still significant discrepancies between different particle size-distribution data sets, especially for the sub-10 nm size range (Kangasluoma et al., 2020). In the sub-10 nm size range, a large fraction of the sample (typically > 95 %) is lost in the measurement system due to diffusional losses, low charging probability, and low detection efficiency of the CPC especially in the sub-5 nm size range, emphasizing the need to acquire sufficient statistics for the counted particles.

The number of registered counts in the CPC is determined from the total size-dependent penetration of the DMPS/SMPS, the CPC aerosol flow rate through the optics, and the sampling interval for an individual size. Most recent advances in the sub-10 nm size distribution instrumentation have been focused on increasing the sampling time (Stolzenburg et al., 2017), size resolution (Kangasluoma et al., 2018), or inversion performance (Stolzenburg et al., 2022a). However, large advances are expected simply by using a CPC with a large aerosol flow rate, which linearly increases the number of counted particles. In addition, it remains unquantified to what extent improved counting statistics provide more reliable results on quantities typically inferred from sub-10 nm size distributions, such as the particle growth and formation rate. Solid uncertainty estimates for these size-distribution-derived quantities are rare (Dada et al., 2020; Kangasluoma and Kontkanen, 2017) or only provided via sophisticated inversion schemes (Ozon et al., 2021).

In the current work, we use a new laminar flow CPC (modified Airmodus A20)
that has 2.5 L min^{−1} aerosol (and optics) flow rate within a DMPS
(Kangasluoma et al., 2015). It was operated in
Hyytiälä, Finland, in parallel with a TSI 3776 as the detector
downstream of the same DMPS system (raw particle number size distributions are provided in Stolzenburg et al., 2023). Here, we demonstrate the improved data
quality given by the larger counting statistics, perform an uncertainty
analysis for the system, and finally determine the effect of the counting
statistics on the calculations of the particle growth and formation rate
through Monte Carlo analysis (Monte Carlo analysis software code is provided in Stolzenburg and Laurila, 2023).

## 2.1 Measurement setup

The measurements were performed from 24 March–19 May 2017 at the SMEAR II
station (Station for Measuring Ecosystem–Atmosphere Relations;
Hari and Kulmala, 2005) The station is located in Hyytiälä, southern
Finland (61^{∘}51^{′} N, 24^{∘}17^{′} E). The DMPS system used in this measurement has a short Hauke-type DMA that was used to select particle sizes in the range of 1–40 (Aalto et al., 2001)
and operated at an aerosol-to-sheath flow ratio of 4 L min^{−1} $/$ 20 L min^{−1}. The
modified Airmodus A20 CPC and the TSI 3776 CPC measured in parallel in the
DMPS system, as illustrated in Fig. 1. In parallel to this nano-DMPS setup, a
long-DMPS using a different DMA (long-column Hauke DMA) but the same inlet
and charger was operated simultaneously (Aalto et al.,
2001). In addition, also the total aerosol number concentration above 4 nm
is determined using a TSI 3775 CPC sampling outside air without an upstream
DMA.

## 2.2 CPCs

The modified Airmodus A20 CPC is a laminar flow CPC, where the entire sample
flow is heated and saturated with butanol. The saturated sample flow goes to
a multi-tube (six tubes) condenser, where the temperature is decreased to
activate the aerosol particle growth by condensation, followed by optical
detection. The nominal cut-off diameter, using the factory settings, of the
Airmodus A20 CPC is 7 nm. The TSI 3776 CPC is also a laminar-type CPC, but
in contrast to the A20 CPC, the TSI 3776 CPC utilizes the ultrafine CPC
design, where the sample flow is introduced in the middle of the condenser
with a capillary (Stolzenburg and McMurry, 1991). The TSI 3776
CPC was operated with the high-flow setting, where the CPC draws an inlet
flow of 1.5 L min^{−1}, of which 1.2 L min^{−1} is directed to a bypass. Of the remaining 0.3 L min^{−1}, 0.25 L min^{−1} is used as a sheath flow and 0.05 L min^{−1} as the sample flow;
i.e. the effective detector flow of undiluted sample in the CPC optics is
only 0.05 L min^{−1}.

Kangasluoma et al. (2015) showed that a
conventional, unsheathed CPC can be tuned for even sub-3 nm particle
detection by increasing the temperature difference between the saturator and
the condenser and by adjusting the inlet flow rate. With the factory
settings of the Airmodus A20 CPC, the saturator temperature is
39 ^{∘}C, and the condenser temperature is 15 ^{∘}C. The modified Airmodus A20 CPC used in this study has a saturator
temperature of 44 ^{∘}C and a condenser temperature of 10 ^{∘}C, and the inlet flow rate was increased from 1 to 2.5 L min^{−1}, which is entirely analysed in the optics unit, resulting in a
factor of 50 difference in analysed sample flow between the two CPCs. While
higher detector flow rates would result in even better counting statistics,
it would require adjustments in the CPC design to achieve similar particle
activation due to lower supersaturations and would also result in a lower
size resolution for the DMPS system if the sheath flow rate remains constant
(higher sheath flow rates would in turn reduce the dynamic size range of the
DMPS).

The detection efficiency of the CPCs was characterized using negative silver
particles produced with a tube furnace. The test particles were charged with
a ^{241}Am radioactive source and size classified with a short Hauke DMA
running at aerosol flow rate of 4 L min^{−1} and sheath flow rate of 20 L min^{−1}. The
CPCs, TSI 3776, modified A20, and standard A20 were calibrated one by
one against a TSI electrometer 3068B running at 1 L min^{−1} flow rate.

Figure S1 in the Supplement shows the cut-off calibration curves for the CPCs. The 50 % cut-off diameters of the Airmodus A20, the modified Airmodus A20 and the TSI 3776 CPC are approximately 5.5, 2.9, and 2.0 nm, respectively. With the modifications, the modified Airmodus A20 CPC has a performance almost comparable to the TSI 3776 CPC. It should be noted that this specific device in this specific calibration performed exceptionally well, as its nominal cut-off is typically closer to 2.5–3 nm for silver test particles (Wlasits et al., 2020).

Apart from their differences in activation efficiency and effective detector flow rate, the two CPCs have different response times to a change in aerosol concentration, which are ∼ 0.1 s for the TSI 3776 and ∼ 1 s for the (unmodified) Airmodus A20 (Enroth et al., 2018). However, as we will see below, that small difference does not affect our approach in comparing the counting statistics of the two CPCs.

## 2.3 Counting process of a CPC: Poisson process

A random variable *N* has a Poisson distribution with the parameter *μ**τ*>0, where *τ* is the measurement time, and *μ* is the
intensity (rate) of the process, if the random variable can obtain discrete
values (0, 1, 2, 3, …) within the time interval *τ*. If the
process is characterized by the following properties, (1) for *τ*=0 we
have *N*(0)=0; (2) in separate time intervals, the numbers of
detected events are independent of each other; and (3) the number of events in
any interval of length *τ* obey the Poisson distribution:

A Poisson distribution can be shown to have the following properties: the
expected value *E*[*N*] of the distribution can be calculated as
*E*[*N*]=*μ**τ*, and the standard deviation (*σ*)
can be calculated as $\mathit{\sigma}=\sqrt{\mathrm{VAR}\left[N\right]}=\sqrt{\mathit{\mu}\mathit{\tau}}=\sqrt{E\left[N\right]}$.

In a CPC, the particles are counted in the optical unit of the CPC, where a
nozzle directs the particle stream to cross a laser beam perpendicularly.
Light is scattered from the laser beam as the particles cross it, and the
scattered light is collected by a photodiode. In typical optics with
∼ 1 L min^{−1} aerosol flow, the probability of coincidence in the
counting process is negligible with moderate number concentrations
(< 30 000 cm^{−3}), which are typically measured
downstream of a DMPS system. In our DMPS, the voltage is stepped from 3 to
1000 V in 17 steps (corresponding to selected mobility diameters of 2.07 to
40 nm assuming singly charged particles), with a settling time of 1 s
at the beginning of each voltage step (which should remove any bias from
different response times of CPCs, if they are ≤ 1 s). The measured
particle number concentration *C* (in cm^{−3}) for size is
determined by the number of particles *N* counted in the time interval *τ* where the voltage is kept constant (which varies between 3.5 s for
the largest size and 64 s for the smallest size) by using the
volumetric flow rate through the optics *Q*_{opt}:

If we assume that the number concentration remains constant during the voltage scan of the DMPS (which is anyway also a requirement for any inversion procedure which considers multiply charged aerosols), the counting process in the DMPS can be considered a Poisson process.

In our setup, we can neglect the total penetration of the system since the
compared CPCs measure in parallel in the same DMPS system, and the total
penetration is the same for both. This allows us to compare the raw data
from the CPCs without an inversion and the uncertainties related to it
(Stolzenburg et al., 2022a). As our DMPS
outputs the average concentration during each voltage step, we need to
rearrange Eq. (2) for the counted particles *N*. This also shows that we can
predict that a factor 50 increase of *Q*_{opt} (effective
undiluted optics flow of 0.05 L min^{−1} in the TSI 3776 versus 2.5 L min^{−1} in the modified Airmodus A20) should lead to a factor 50 increase of *N*:

## 2.4 Uncertainty in CPC measurements

Uncertainty is a fundamental concept in statistics and probability, and it
occurs in all measurements. The uncertainty of a measurement can be
systematic, due to human error or resulting from the natural fluctuation of
the observed system. In most cases, the total uncertainty of the measurement
is a combination of uncertainty from multiple sources.
Ultimately, we are interested in the uncertainty of the data obtained from
an individual CPC within a DMPS setup, which could be used within
uncertainty estimates of subsequently derived variables (*J*, GR). However, we are typically not able to quantify that total uncertainty and are not able
to disentangle the counting process from other sources of uncertainty, such as
electronic noise or flow variations in the CPC optics (called measurement
error in the following). However, our specific setup allows us to confine
the counting uncertainty due to the availability of another CPC.

We chose the following approach to obtain an uncertainty estimate of the
measurements with the DMPS using the TSI 3776 as a detector. First, only
data for particles ≥ 6 nm are used for the error analysis to
ensure that the detection efficiencies of the CPCs do not affect the result.
As we can see in Fig. S1, at 6 nm, the calibration curves of both CPCs have
plateaued. Next, we choose the measurement time where the modified Airmodus A20 measures particle counts *N* in certain narrow ranges [*N*_{1},
*N*_{2}], where ${N}_{\mathrm{2}}=\mathrm{1.05}\cdot {N}_{\mathrm{1}}$ with ${N}_{\mathrm{1}}\le N\le {N}_{\mathrm{2}}$. The counts from the
corresponding times are then selected from the parallel measuring TSI 3776.
These selected particle counts are plotted as a normalized histogram, and a
Gaussian probability density function (PDF) is fitted to the data (which is
a good approximation to a Poisson distribution when *E*[*N*]>10). This
approach of choosing finite count intervals from the Airmodus A20 data
instead of just using a single count value is due to the otherwise limited
statistics which would not allow for solid fits of the corresponding count
distributions of the TSI 3776. Figure 2 shows four examples of the resulting
histograms and fits.

We are now interested in the uncertainties determining the width of these
PDFs. By selecting count ranges in the modified Airmodus A20, we select
measurements with an actual number concentration *C*_{true}±Δ*C*_{true}, where the uncertainty originates from the counting and
measurement error in the modified Airmodus A20 and the finite width of
selected counts in the interval range (with the relative error due to this
kept below 5 % by our interval selection ${N}_{\mathrm{2}}=\mathrm{1.05}\cdot {N}_{\mathrm{1}}$), which are assumed to be independent error sources and hence can
be expressed in relative uncertainties as follows:

The *C*_{true} constrained by the selection of modified Airmodus A20
measurements is also measured simultaneously by the TSI 3776. Therefore,
the PDF of counts measured in the TSI 3776 (or its width, i.e. its relative
uncertainty $\mathrm{\Delta}{N}_{\mathrm{TSI}}^{\mathrm{PDF}}/{N}_{\mathrm{TSI}}$) results from the uncertainty in the *C*_{true} values selected by the modified Airmodus A20
measurements, the counting error of the 3776, and the measurement error of
the 3776, expressed in relative uncertainties as follows:

We see that besides the measurement errors, we can specify all terms in Eq. (5). As we aim to determine the total error of the TSI 3776 of an
independent measurement of a concentration *C*_{true} given by the
uncertainties in counting and measurement, which we can now link to the
measured width of the PDF via Eq. (5) obtaining

If we now neglect the uncertainty in the measurement of the modified Airmodus A20 CPC, Eq. (6) provides an upper estimate of the total error in the CPC 3776.

## 2.5 Growth rate and formation rate and propagated uncertainties via MC simulations

Using these error estimates, we can derive the corresponding uncertainties
in the quantities typically derived from DMPS size-distribution data, the
growth rate (GR) and formation rate (*J*). Here, we calculate the GR using the 50 % appearance time method (Stolzenburg et al., 2018;
Lehtipalo et al., 2014) with an automated algorithm, which after manually
defining a time window for the NPF event, fits sigmoidal functions to the
rise of the measured raw number concentration (the approach is independent
of the absolute magnitude of the signal and hence the inversion procedure;
see Lehtipalo et al., 2014) in each size channel separately. The 50 %
appearance times are then plotted against the sizes of the corresponding
channels, and a linear interpolation is used for the size range 3–6 nm
(2.99–6.28 nm) and 6–10 nm (6.28–10.94 nm) to obtain GR_{3–6} and GR_{6–10} as the slope of that interpolation, respectively.

The formation rate can be calculated for particle size range [*d*_{p},
*d*_{p}+Δ*d*_{p}] according to Eq. (7) (Kulmala
et al., 2012):

Here, CoagS is the coagulation sink (loss rate of particles in that size
range with the background particles due to coagulation), and ${N}_{{d}_{\mathrm{p}}}$ is the
number concentration of the particles in the size range [*d*_{p},
*d*_{p}+Δ*d*_{p}]. The coagulation sink is calculated for
the geometric mean diameter of the selected size range and in the atmospheric
conditions typical for the SMEAR II; it can be empirically estimated from
the condensation sink (CS) of a non-volatile vapour
(Dal Maso et al., 2005) as in Eq. (8) (Lehtinen et al., 2007):

We use the size interval [3 nm, 6 nm] to calculate the formation rate at 3 nm (*J*_{3}) in all subsequent calculations. For the automated algorithm,
the integrated concentration ${N}_{{d}_{\mathrm{p}}}$ of the interval was smoothed, and the GR_{3–6} value for the specific NPF day was used as input for the last
term. The diurnal variation of *J*_{3} was then fitted by a Gaussian
expression, and its peak value was used as the NPF-event-specific *J*_{3}
value.

We performed a Monte Carlo simulation on one of the NPF days (28 March 2017). New sets of data were generated from the original data 10 000 times, by altering the measured counts in each size channel for each
measurement time according to their underlying uncertainties. We performed
three sets of MC simulations. First and second, we use a Poisson counting
error to vary the TSI 3776 and the modified Airmodus A20 data (assuming a
$\sqrt{N}$ uncertainty). The generated input data (counts) were used to
directly calculate GR_{3–6} and GR_{6–10} as the appearance time
method can be performed on the raw signal. For the calculation of the
formation rate, we inverted the raw signal into a size distribution using a
least-squares algorithm which also considers the data above 10 nm obtained
from the long-DMPS. Comparison of the resulting formation and growth rates
allows the investigation of the effect of increasing counting statistics
with respect to these size-distribution-derived quantities. As a third
simulation, we assume the total error for the TSI 3776 derived via Eq. (6)
(upper error estimate) as the input uncertainty in the Monte Carlo runs
altering the raw counts and compare it with the Poisson-only case of the TSI 3776 to investigate the magnitudes of counting and measurement error on
GR_{3–6}, GR_{6–10}, and *J*_{3}. The relative uncertainties for each
size-distribution evolution measurement (in time and size) used as input for
all three Monte Carlo simulations are shown in Fig. S2 in the Supplement.

## 3.1 Effect of counting statistics on the inverted size distributions and number closure

We analysed the dataset by classification of the NPF event days (Dal Maso et
al., 2005) and calculated formation and growth rates for the subset of
class-I NPF event days. Figure 3 shows an example NPF day (28 March
2017) from both CPCs (modified Airmodus A20 Fig. 3a and TSI 3776 Fig. 3b).
The 28 March is chosen as the example day as it is a typical class-1
NPF event day with a strong nucleation rate but not much higher than
average GR, such that the nucleation mode persists over a long enough time in
the sub-10 nm range to investigate the effect of improved counting
statistics in full detail. We can see that the modified Airmodus A20
produces a smoother distribution in the areas of low concentrations
(blue-to-yellow colour range). Besides the lower nominal cut-off in the
laboratory calibration of the TSI 3776 (Fig. S1), the signal at the small
sizes below 5 nm is noisier in the TSI 3776-derived size distribution
compared to the modified Airmodus A20-derived size distribution.
Potentially, the overall reduced statistics counterbalance the effect of a
more efficient detection at these sizes. Moreover, it needs to be noted that
ambient cut-offs are subject to larger uncertainties due to the unknown
chemical composition of the counted particles and the composition-dependent
response of the CPCs, which can be more than 3 nm difference for the
*d*_{50} cut-off diameter between different seed materials for the unmodified
Airmodus A20 (and only 1.2 nm maximum variation for the TSI 3776)
(Wlasits et al., 2020).

Next, we compare the performance of the DMPS using different detectors with respect to the number closure with a simultaneously measuring total CPC (TSI Model 3775, nominal cut-off 4 nm). The correlation of the full campaign dataset between the integrated number concentration of the DMPS system (above 4 nm) and the total concentration measurement with the CPC 3775 is shown in Fig. 4 for both detectors (Fig. 4a using the TSI 3776 in the inversion and subsequent integration and Fig. 4b using the modified Airmodus A20). Pearson's coefficient of correlation is high for both (0.992 and 0.994) but slightly better in cases when the modified Airmodus A20 is used within the DMPS inversion, which is reasonable due to the increased statistics. However, the data deviate from the 1 : 1 relation (0.89 slope for the modified Airmodus A20, which is more significant than for the TSI 3776 based DMPS data with a slope of 0.94). This could be due to a different plateau value reached in the counting efficiency curves and not correctly accounted for by the calibration. Wlasits et al. (2020) showed that plateau values of the same instrument vary slightly between different calibrations. Therefore, this could easily lead to offsets in the inversion, resulting in the observed discrepancies in the total number concentration.

## 3.2 The effect of increased counting statistics on the particle formation and growth rates

In Fig. 5, we compare the calculated GR_{3–6}, GR_{6–10}, and *J*_{3} values obtained from the DMPS data with the different underlying detectors for all NPF class-I events (see Dal Maso
et al., 2005) recorded throughout the campaign (in total 19 events). We
observe strong correlations in the derived growth and formation rates, with
the lowest correlation coefficient for GR_{3–6}, where the signal is most noisy. Interestingly, the formation rate is more robust, even if derived at 3 nm, where also the GR_{3–6} is used within the calculation of Eq. (7).
However, as shown in Fig. 4, the modified Airmodus A20 measured slightly
lower concentrations compared to the TSI 3776, while GR_{3–6} was
measured higher by the Airmodus A20 for values above 3 nm h^{−1}.
Therefore, in these cases with a high growth term $\left(\frac{\mathrm{GR}}{\mathrm{\Delta}{d}_{\mathrm{p}}}{N}_{{d}_{\mathrm{p}}}\right)$ possibly dominating the formation rate calculations due to a fast
growth rate (> 3 nm h^{−1}), the lower *N*_{3–6} might
compensate for the higher GR_{3–6} reducing the fluctuations between the two instruments. In addition, the other terms $\left(\right)open="(">\frac{\mathrm{d}{N}_{{d}_{\mathrm{p}}}}{\mathrm{d}t}$ and
$\left({\mathrm{CoagS}}_{{d}_{\mathrm{p}}}{N}_{{d}_{\mathrm{p}}}\right)$ in Eq. (7) might also buffer the higher GR due to *N*_{3–6} values in that case.

In Fig. 6 we present the results from our Monte Carlo analysis of
28 March 2017, comparing the performance of the modified Airmodus A20
with the TSI3776, assuming the measured signal is only subject to a counting
uncertainty. Figure 6a and b present the results of the 10 000 GR_{3–6} and GR_{6–10} calculations performed with the same automated appearance time
algorithm, showing the obtained 50 % appearance times at each diameter
(channel) on top of the original size distribution and the corresponding
linear fits for the GR estimate. Apparently, the smaller the channel size,
the larger the spread between the appearance time results, especially for
the TSI 3776, where the relative uncertainty of each measurement becomes
very large below 4 nm due to the limited count rates (which is in the range
of 10 counts per measurement during NPF; see also Fig. S2 in the
Supplement). It needs to be noted that it seems to be especially the channel
at 3 nm, which has a broad spread in 50 % appearance times dominating the variation in the subsequent GR_{3–6} derivation.

This directly translates into the significantly larger variance of the
GR_{3–6} values derived from the TSI 3776 compared to the modified
Airmodus A20 (Fig. 6d and e). For GR_{3–6} the relative statistical
uncertainty (defined as 1*σ* standard deviation divided by the initial
GR_{3–6} result obtained from the actual measurement data) from the
counting error is much larger for the TSI 3776 ($(\mathrm{\Delta}{\mathrm{GR}}_{\mathrm{3}\text{\u2013}\mathrm{6}}/{\mathrm{GR}}_{\mathrm{3}\text{\u2013}\mathrm{6}}{)}_{\mathrm{TSI}}^{\mathrm{count}}$ ∼ 16 %) compared to the modified Airmodus A20 ($(\mathrm{\Delta}{\mathrm{GR}}_{\mathrm{3}\text{\u2013}\mathrm{6}}/{\mathrm{GR}}_{\mathrm{3}\text{\u2013}\mathrm{6}}{)}_{\mathrm{A}\mathrm{20}}^{\mathrm{count}}$ ∼ 1 %). GR_{6–10} shows lower overall uncertainties and fewer, but still significant, differences between the two CPCs (2 % compared to 0.3 %). Interestingly, the mean of the Monte Carlo
distributions is slightly offset between the two CPCs for both GR_{3–6} and GR_{6–10}, demonstrating the observed variations shown in Fig. 5
and with the mean of the distributions roughly centred around the original
result. However, even though we saw good correlation for the *J*_{3} values
within the campaign derived from both instruments, it seems that *J*_{3} is
also heavily influenced by the counting statistics. In Hyytiälä, the
most dominant term in the calculation of the formation rate is often the
growth term out of the bin of interest, i.e. $\frac{\mathrm{GR}}{\mathrm{\Delta}{d}_{\mathrm{p}}}{N}_{{d}_{\mathrm{p}}}$
(Eq. 7 and Fig. 6c), especially at fast growth rates, which is confirmed
here. Therefore, the fluctuations in GR_{3–6} are directly translated (Fig. 6f) into large uncertainties for the TSI 3776-derived *J*_{3} ($(\mathrm{\Delta}{J}_{\mathrm{3}}/{J}_{\mathrm{3}}{)}_{\mathrm{TSI}}^{\mathrm{count}}$ ∼ 13 % relative uncertainty) and much lower in the modified Airmodus A20-derived *J*_{3} ($(\mathrm{\Delta}{J}_{\mathrm{3}}/{J}_{\mathrm{3}}{)}_{\mathrm{A}\mathrm{20}}^{\mathrm{count}}$ ∼ 1 %).

In addition, it needs to be noted that 28 March 2017 was one of
the days with the highest formation rate (*J*_{3} ∼ 1.5 cm^{−3} s^{−1}) throughout the campaign. Therefore, we repeated the analysis for 2 additional days with significantly lower *J*_{3} (5 and 6 May 2017, with *J*_{3} = 0.05 cm^{−3} s^{−1} and *J*_{3} = 0.15 cm^{−3} s^{−1}, respectively). We present the Monte Carlo results for GR_{3–6} and *J*_{3} for the intermediate formation rate day (6 May 2017) in Fig. S3 in the Supplement and show all results for GR_{3–6} and *J*_{3} in Table 1. As expected, the lower *J*_{3} also resulted in lower count rates in both CPCs during NPF. Therefore, also a larger counting uncertainty in the size-distribution-derived quantities was observed, with up to 23 % relative uncertainty in GR_{3–6} and 16 % in
*J*_{3} when the TSI 3776 is used and with a still significant reduction for
the modified Airmodus A20 down to ∼ 9 % relative uncertainty
(for the 6 May 2017). At very low *J*_{3} (5 May 2017, Fig. 7), the Monte Carlo distributions for the TSI 3776 data get skewed (with the
mean of the distribution also deviating significantly from the original
result), and the Monte Carlo results show a bimodal distribution, with
unphysical GR values around 0, indicating problems with the automated GR
fitting. The relative uncertainty becomes as large as 40 %. This shows
that GR values derived at such low number concentrations and with such low
counting statistics are not reliable. Only instrumentation which provides
enough signal can be used: even though the modified Airmodus A20 relative
uncertainty already becomes as large as 10 %, this value is still lower
than the relative uncertainty of GR_{3–6} for the TSI dataset of a very
strong NPF event day with *J*_{3} almost 2 orders of magnitude higher.
Altogether, the counting uncertainties derived for all 3 d analysed
by the Monte Carlo approach can explain the observed scatter between the
values derived by the two instruments (see error bars for the three selected
events in Fig. 5), which implies that the counting uncertainty is a major
issue when GR and *J* values are compared between different instruments.

## 3.3 Estimating the total error of the TSI 3776 and its effect on the particle formation and growth rates

We now aim to estimate the total error in a CPC measurement based on our dual setup. As described by Eq. (6), we can obtain an upper estimate of the total error in the TSI 3776 measurement by selecting small count ranges in the modified Airmodus A20 and estimating the width of the resulting count distribution in the TSI 3776 at simultaneous measurements. In Fig. 8a we show the upper relative error estimate together with the pure counting error ($\mathit{\sigma}=\sqrt{N}$) for a set of selected count intervals in the modified Airmodus A20 versus the expected value of counts in the TSI 3776 ($E\left[{N}_{\mathrm{TSI}}\right]={Q}_{\mathrm{TSI}}/{Q}_{\mathrm{A}\mathrm{20}}\cdot {N}_{\mathrm{A}\mathrm{20}}$). We see that the relative uncertainty is significantly larger than what would be expected from a pure counting error, indicating that there are also other important sources of uncertainty in a typical DMPS measurement, resulting from fluctuations in flow rates or electronic noise. If we further assume that the relative uncertainty of such an additional source is the same for any CPC, we can further simplify Eq. (6) by setting ${\left(\frac{\mathrm{\Delta}{N}_{\mathrm{CPC}}^{\mathrm{meas}}}{{N}_{\mathrm{CPC}}}\right)}^{\mathrm{2}}={\left(\frac{\mathrm{\Delta}{N}_{\mathrm{TSI}}^{\mathrm{meas}}}{{N}_{\mathrm{TSI}}}\right)}^{\mathrm{2}}={\left(\frac{\mathrm{\Delta}{N}_{\mathrm{A}\mathrm{20}}^{\mathrm{meas}}}{{N}_{\mathrm{A}\mathrm{20}}}\right)}^{\mathrm{2}}$ and even solve it for that missing error source, which is shown in Fig. 8b. We obtain a roughly constant value of around 4 % across all count ranges, also indicating that these fluctuations are indeed independent from the counting error.

To estimate the influence of such additional uncertainties in CPC
measurements on the size-distribution-derived quantities GR_{3–6},
GR_{6–10}, and *J*_{3}, we performed another Monte Carlo simulation using a fitted expression as in Eq. (6) (counting uncertainty plus an additional
measurement uncertainty, where its relative magnitude is the free parameter
of the fit) to the total error in Fig. 8a as the input for the variation of
the measured counts in the TSI 3776. Figure 9 shows the resulting histograms
for GR_{3–6}, GR_{6–10}, and *J*_{3} for the strong NPF event day together with the results from the Monte Carlo analysis using the pure counting uncertainty only. While the
distributions are even further skewed, the relative widths do not
dramatically increase further. For the events at reduced *J*_{3} (Fig. S3 in
the Supplement and Table 1), the influence of the measurement error on the
size-distribution-derived quantities GR_{3–6} and *J*_{3} becomes almost
negligible compared to the even higher counting uncertainties as almost no
further broadening of the result distributions is observed. Altogether,
this clearly demonstrates that the counting uncertainty is the dominant
source of error for nucleation and growth rate determination when a TSI 3776
ultrafine CPC is used.

Our limited dataset does not allow for the reverse procedure due to a lack of statistics (i.e. selecting narrow count ranges in the TSI 3776 and obtaining the PDF for the simultaneous measurements of the modified Airmodus A20), and hence we do not provide a detailed Monte Carlo analysis on the effects on the growth and formation rate. However, as the relative counting error is so much lower in the modified Airmodus A20, we suspect that this additional source of uncertainty would dominate the formation and growth uncertainties in that case by the following simple reasoning: the relative counting uncertainty scales with $\mathrm{1}/\sqrt{N}$, and the measurement uncertainty seems to be independent of the number of counts (Fig. 8b), and hence the ∼ 4 % measurement uncertainty start to dominate the total uncertainty above 625 counts as $\mathrm{1}/\sqrt{\mathrm{625}}=\mathrm{0.04}$, which is roughly the sub-5 nm count rates measured in the modified Airmodus A20 during the NPF event of 28 March 2017.

The strength and importance of NPF with respect to the climate system is often characterized by formation and growth rates, which are commonly derived from the evolution of measured particle number size distributions obtained from DMPS/SMPS systems. However, the uncertainties in the DMPS measurements and their effect on the size-distribution-derived quantities are not well quantified. As the CPC counting process can be considered a Poisson process, the resulting uncertainty from the counting process can be non-negligible at the low count rates and might dominate the uncertainty in the derived size distribution and formation and growth rates.

Here, we deploy a DMPS system with a modified Airmodus A20 CPC providing a
factor 50 higher counting statistics compared to the commonly used TSI 3776
ultrafine CPC. We found that the modified Airmodus A20 provides smoother
number size distributions, especially in the case of low concentrations of
ultrafine particles and achieves very good correlation with simultaneous
absolute number concentration measurements. The difference between the
counting statistics of the CPCs is propagated to the values derived from the
measured number size distribution, resulting in significantly reduced
uncertainties for GR_{3–6} (1 % compared to 16 %), GR_{6–10} (0.3 % compared to 2 %), and *J*_{3} (1 % compared to 13 %). This
effect is even stronger, when the formation rates and hence number
concentrations are low, where a reliable GR estimate might only be possible
with a DMPS with sufficient counting statistics. In addition, our dual CPC–DMPS setup allowed for a quantification of the total uncertainty related to
the CPC measurement in a DMPS system, showing that additional sources of
uncertainties with a relative uncertainty of around 4 % are present at all count rates. However, we showed that the counting uncertainty is the main source of error for the size-distribution-derived quantities *J* and GR for the widely used TSI 3776. The additional sources of uncertainty might only
become important in the derivation of the nucleation and growth rates when
the counting uncertainties are reduced as in the case of the modified
Airmodus A20.

This study shows significant improvement in the determination of the formation and growth rate during NPF by the deployment of a DMPS with improved counting statistics. The wide deployment of such instrumentation which is optimized for sub-10 nm measurements could significantly reduce our uncertainties in formation and growth rate determination or even allow for the application of better analysis tools due to the increased statistics (Pichelstorfer et al., 2018; Ozon et al., 2021) and hence boost our understanding of NPF; for example, they provide better mass closure in aerosol growth (Stolzenburg et al., 2022b). However, this study also shows that other sources of uncertainty are typically present in DMPS measurements, which also need to be understood and potentially be reduced or at least be well quantified, which requires future work on CPC techniques.

The software code for performing the Monte Carlo analysis is available under https://doi.org/10.5281/zenodo.7962563 (Stolzenburg and Laurila, 2023).

Raw particle number size distribution data and retrieved growth and formation rates are available under https://doi.org/10.5281/zenodo.7962336 (Stolzenburg et al., 2023).

The supplement related to this article is available online at: https://doi.org/10.5194/amt-16-2471-2023-supplement.

TL, PA, JV, and JK performed the measurements; DS and TL analysed the data and performed the simulations; DS, TL, TP, and JK were involved in the scientific discussion and interpretation of the results; DS and TL wrote the manuscript; and all co-authors commented on the manuscript.

Joonas Vanhanen is the Chief Technology Officer of Airmodus Ltd., the company producing and selling the A20 CPC. The remaining authors have no conflicts of interest to declare. This study was independently performed and was not co-funded by Airmodus Ltd.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank Lubna Dada for her support in nucleation rate calculations.

This work was funded by the Academy of Finland Flagship via the Atmosphere and Climate Competence Center (ACCC; grant no. 337549) and the Academy of Finland (grant nos. 1325656, 346370, and 79999129). It also received funding from the University of Helsinki 3-year grant (grant no. 75284132) and the University of Helsinki ACTRIS-HY. It also received support from the European Union's Horizon 2020 Research and Innovation programme under a Marie Skłodowska–Curie Action (grant agreement no. 895875) (NPF-PANDA), from the European Commission through Research Infrastructures Services Reinforcing Air Quality Monitoring Capacities in European Urban & Industrial AreaS (RI-URBANS; grant no. 101036245) and through ACTRIS-CF (329274) and ACTRIS-Suomi (328616).

Open-access funding was provided by the Helsinki University Library.

This paper was edited by Hang Su and reviewed by two anonymous referees.

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