The differential mobility particle sizer (DMPS) is designed for measurements
of particle number size distributions. It performs a number of measurements
while scanning over different particle sizes. A standard assumption in the
data-processing (inversion) algorithm is that the size distribution remains
the same throughout each scan. For a DMPS deployed in an urban area this
assumption is likely to be violated most of the time, and the resulting size
distribution data are unreliable. To improve the reliability, we developed a
new algorithm using a statistical model in which the problematic assumption
was replaced with more realistic smoothness assumptions, which were expressed
through Gaussian process prior probabilities. We tested the model with data
from a twin DMPS located at an urban background site in Helsinki and found
that it provides size distribution data which are much more realistic.
Furthermore, particle number concentrations extracted from the DMPS data were
compared with data from a condensation particle counter. At 10

There is no direct way of measuring the size distribution of fine particles.
To get information on the size distribution, mobility particle size
spectrometers

A DMPS typically performs a few tens of measurements while scanning over
a wide range of electrical mobilities. Each measurement takes several
seconds, and a typical waiting time between the measurements is around 10

Size distributions from 2 March 2015 according to the old inversion algorithm, which assumes stationary particle size distributions during each scan.

Three particle number size distributions from 2 March 2015 according to the old inversion algorithm, which assumes stationary particle size distributions during each scan.

A few studies have addressed this issue of possible size distribution changes
happening during a scan.

In this work, we developed a new inversion algorithm for processing DMPS data
from locations with fluctuating particle number concentrations. The particle
number size distribution was modelled as a function of time and particle size
using a Gaussian process (GP) model

Particle size distributions are usually described by the

The DMPS comprises a neutraliser (bi-polar charger), a differential mobility
analyser (DMA), and a condensation particle counter (CPC). In the
neutraliser, ionising radiation ensures that the particles in the sampled air
reach the equilibrium charge distribution. This charge distribution is known
and depends on the particle size. In the DMA, the voltage and airflow are
adjusted to select particles with a certain electrical mobility

The transfer function

A GP, or Gaussian random field, is a stochastic process that can be used to
define probability distributions over functions, and it is a generalisation
of the multivariate normal (Gaussian) distribution

Let us define a latent function

Each measurement

The particle number size distribution is assumed to be a smooth function of
the particle size, and it is assumed to vary smoothly over time. These
properties are modelled by giving a GP prior for the latent function

We assume that the mean function is constant,

The prior variance of mean

We used data from the urban background station SMEAR III

Neighbourhood of the SMEAR III station.

We got the transfer function from the old inversion algorithm used at
University of Helsinki. For each measurement

In our pre-processing of the data we also had to reconstruct the particle counts in all measurements by multiplying the saved concentrations, sample flows, and durations of measurements. We rounded the results of this multiplication to get integer counts. This reconstruction may be affected by rounding errors, which, however, are of secondary importance.

We processed data from 26 February to 7 March 2015 in batches of eight scans.
After fitting the model to the data, for the post-processing we defined
a grid with 5

We did all calculations on a normal desktop computer. For each batch the model fitting took about 2 min, and another 2 min were spent on the post-processing. The sampling of size distributions was the most time-consuming part of the post-processing.

Training inputs for the scan on 2 March between 11:00 and
11:10 UTC

Independent measurements of particle number concentrations were obtained with
a CPC (TSI 3787 water CPC), which detected particles larger than
5

Given the model description and data, we approximate the posterior
distribution

The hyperparameters,

It is possible to analytically solve the gradients of

After finding

We will evaluate the results from our algorithm both by looking at some illustrative examples and by comparing resulting particle number concentrations for the whole period with CPC data.

As a first test of the algorithm, let us consider periods without
fluctuations, meaning periods for which the old algorithm performed well. As
expected, for these periods our results agree well with the results from the
old algorithm. The only clear difference is that we obtain smoother size
distributions with our new algorithm as in the example in
Fig.

The size distribution obtained with the old inversion algorithm and expected size distributions (new inversion) for a period with little fluctuation on 26 February.

Upper panel: expected size distributions on 2 March between 10:30
and 12:00 UTC

Posterior variance of

In our next example (2 March 10:30–12:00 UTC

Particle number concentrations on 2 March between 10:30 and
12:00 UTC

The time evolution of the particle number concentrations obtained with the
DMPS agrees well with the CPC measurements (Fig.

Expected size distribution before, during, and after the peak on
2 March at 11:03 UTC

Expected size distribution with 95

Even less size information is available for the brief concentration peak occurring
between 11:31:25 and 11:32:00 UTC

Histogram of correlations between 30

To evaluate the performance for the processed 10-day period, we
compared mean particle number concentrations obtained from the DMPS and the
CPC data at 10

Particle number concentrations on 7 March between 10:30 and
11:00 UTC

Let us illustrate this with an example
(Fig.

The results above are based on a few simplifying assumptions. We assumed that
the particle concentration only changes a little during each measurement.
This is not necessarily always the case, but the approximation in
Eq. (

In summary, our algorithm extracts well the time evolution of the particle number concentration from the available DMPS data, and in the absence of fluctuations the obtained size distributions fit well with results from the old algorithm. During fluctuations, only little information about the particle sizes is available, and the uncertainties of the size distributions are considerable. Due to a lack of independent size distribution data, a quantitative evaluation of the size distributions obtained for periods with fluctuation was impossible, but there is no doubt that these size distributions are much closer to the truth than the ones obtained with the old algorithm.

In principle, this method should work for the SMPS as well, but we expect the implementation to be more difficult. The continuous scan needs to be divided into a number of counting intervals. If the counting intervals are long, the peaks of the transfer function will be much wider. On the other hand, if the counting intervals are short, the number of training inputs in our model will be high, and our algorithm will be much slower.

We have developed a new algorithm (provided in the Supplement)
based on a Gaussian process model for processing DMPS data, and we tested it
with data from a twin DMPS in an urban background location. Our algorithm
derives

The higher accuracy of the particle number size distributions can benefit studies of aerosols in urban locations and other places with fluctuating size distributions. The higher time resolution is useful, for instance, when attempting to pinpoint sources, given that other data, such as wind observations, exist at a good time resolution. Particle number size distributions at a high time resolution can be obtained with other instruments as well, but this algorithm offers an improvement both for existing and future DMPS data without any need to purchase new hardware.

J. Vanhatalo and N. L. Prisle were funded by the Academy of Finland (grants 266349 and 257411, respectively). Edited by: S. Malinowski