Let us first briefly recall the forward model of a typical MPSS. The forward model is a deterministic mapping from the size distribution, or its discretization, to the expected, noiseless, measurements. The actual measurements will differ from the expectation due to stochastic uncertainties, which we will model later in Sect. . Assume that we have selected I channels to measure, where each channel i is associated with a nominal (i.e., assumed to be singly-charged) particle mobility diameter, which can also be stated as a target electrical mobility Zi that we want to classify, which also determines a set DMA voltage. We will express particle size on a log⁡10-scale, which is the typical convention, and hence define the size distribution function as n(d)=dNdlog⁡10d [cm⁻³ per decade], where N is the cumulative size distribution and d is the particle mobility diameter. For a continuous size distribution, the expected particle counts ỹi measured by the CPC in channel i are: 1ỹi=V∫log⁡10(dmin)log⁡10(dmax)∑pTZ(d,p),Zifp(d)η(d)n(d)dlog⁡10d, where V is the sampled volume, dmin and dmax denote the smallest and largest considered mobility diameters, respectively, TZ(d,p),Zi is the DMA transfer function for channel i where Z(d,p) denotes the electrical mobility of a particle with diameter d and charge p. The transmission and detection efficiencies, including sampling lines and the DMA penetration efficiency as well as the CPC counting efficiency, are represented by η(d), and fp(d) is the charging probability, the probability that a particle of size d carries p elementary charges.

To derive a discrete form of Eq. (), we divide the size range [log⁡10(dmin),log⁡10(dmax)] into M bins, with the jth bin as Δj=[log⁡10(djleft),log⁡10(djright)], denoting its midpoint (geometric mean of the edges) by dj and width by |Δlog⁡10dj|. Further, for simplicity let us assume that in each bin the number concentration n(d), the penetration and detection efficiency η(d), and charging probability fp(d) vary smoothly enough as function of d so that we can approximate them as constant, taking their midpoint values n(dj),η(dj), and fp(dj), which we will denote by nj,ηj, and fp,j. That leaves only the transfer function to be integrated, which we carry out with the trapezoidal rule. Taking all constant terms outside the integral, the discretized version reads 2ỹi≈V∑j=1M∑pfp,j∫ΔjTZ(d,p),Zidlog⁡10dηjnj. Now denote the integral of the transfer function, normalized by the bin width |Δlog⁡10dj|, by 3T̃ij,p=1|Δlog⁡10dj|∫ΔjTZ(d,p),Zidlog⁡10d, define the MPSS kernel, or instrument matrix, for output channel i and size bin j 4Aij=V∑pfp,jT̃ij,pηj, and represent the size distribution using the total particle concentration in a bin Nj:=nj|Δlog⁡10dj|. The forward model can then be written in matrix form as 5ỹ=h(N)=AN, where ỹ=(ỹ1,…,ỹI)T, N=(N1,…,NM)T, and h(⋅) denotes the forward operator that maps the discretized size distribution to the expected CPC counts. The exact forms of the DMA transfer function, the sampling losses, and the counting efficiencies implemented in MPSS-UQ are given in Appendix .

Since the MPSS forward model (Eq. ) depends explicitly on the particle charge distribution, any deviation from the expected distribution can significantly affect the PSD estimates. Let us now investigate the magnitude of the uncertainty in the charge distribution.

A common way to achieve a (in principle known) steady state charge distribution is by bipolar diffusion charging, also known as neutralization. Compared to unipolar charging, bipolar charging has a rather low charging efficiency, especially for small (<10 nm) particles , but the upside is that the charge distribution produced is not as highly dependent on the charging conditions and thus the reproducibility of the charge distribution is better. In bipolar charging, the aerosol is exposed to a high concentration of positive and negative ions which transfer charge to the particles in ion-particle collisions. After a sufficient amount of time the process results in a stationary charge distribution with both positively and negatively charged particles, whose net charge is close to zero but typically slightly negative due to the higher attachment probability of negative vs. positive ions. For sufficient conditions to reach stationarity see for example . Also note that the charge distribution keeps evolving in the tubing downstream of the charger . In this paper, we assume that the concentration of the ions is high enough and the residence time in the charger is long enough to achieve a stationary charge distribution, but we do not model its evolution past the charger.

To model the bipolar charge distribution numerically we use the Fuchs theory as extended and corrected by and later by . We will henceforth refer to this model as the LYF model. The model is based on the limiting sphere approach, where the space around the particles is divided into two zones, an outer diffusional zone and an inner low collisionality zone, which are separated by the limiting sphere. Inside the limiting sphere, particle motion is idealized to be purely ballistic which necessitates the use of correction terms such as three-body trapping, whereas in the diffusional zone the motion follows the principles of continuum mechanics. The LYF model has been shown to be in good agreement with experimental data on particle-ion collision rates in atmospheric conditions , but note that there are other approaches that we could use as well. In the following, we will discuss some input parameters and some computational aspects of our implementation of the LYF model in MPSS-UQ. For a complete theoretical description of the model we refer to and the references therein.

The LYF model characterizes ion–particle collision dynamics through the ion flux coefficient βq,±(d) which represents the rate at which positive or negative ions attach to a particle of diameter d with q elementary charges. For simplicity, we denote the model by FLYF and express this relationship symbolically as 6βq,±(d)=FLYF(d,Z±,m±,T,μ,Mgas,P), where d is the particle diameter, Z± and m± are the ion mobility and mass, and T, μ, Mgas, and P are the temperature, viscosity, mass, and pressure of the carrier gas, respectively. To keep notation light, in the following we suppress the explicit dependence of βq,± and fp on particle diameter d. Note that the ion mobilities themselves depend on the chemical composition of the carrier gas . The carrier gas hence influences the charge fraction both directly via collision dynamics and indirectly via ion mobility. In the sensitivity analysis here, we consider only the effect of varying the ion mobility and compute the corresponding ion masses m± deterministically as shown below, while setting the gas properties to nominal values of T=298 K, μ=18.27×10-6 Pa s, Mgas=28.97 amu, and P=101325 Pa.

The fraction fp of particles carrying p charges can be calculated from the flux coefficients as 7fp=1Ω∏q=1pβq-1,+βq,-n+n-;p≥+11;p=0∏q=p-1βq+1,-βq,+n-n+;p≤-1, where n± denotes the number concentration of positive or negative ions (not to be confused with the particle size distribution n(d)). The normalization constant Ω is given by 8Ω=∑p=-∞-1∏q=p-1βq+1,-βq,+n-n++1+∑p=1∞∏q=1pβq-1,+βq,-n+n-.

Various measurements of the ion mobilities have been carried out over time, using several types of radioactive and X-ray sources. An extensive list of these has been compiled by , and we summarize the reported mobility values in Fig. a. The measured ion mobilities vary over a rather large range, between Z+∈[1.1,1.65] cm2V-1s-1 and Z-∈[1.15,2.10] cm2V-1s-1 with the condition Z-≥Z+. Further, the ion source does not seem to correlate with the mobility values. We therefore define as the plausible range of ion mobilities in an MPSS measurement to consider a trapezoid-like shape around the measured values, denoted by the shaded area in Fig. a. Note that the mobilities Z+=1.35 cm2V-1s-1 and Z-=1.60 cm2V-1s-1 used in the Wiedensohler approximation are approximately in the middle of this trapezoid.

There are considerably fewer studies on the ion masses, and the relationship between the ion mobility and mass also depends on the species of the ion. The ion mass could in principle be inferred from the Stokes-Millikan relation , and there are also more detailed scattering-based mobility models applicable to ions in diatomic gases , but in this work we use the empirical relationship by which is a fit to the experimental data of obtained for a variety of gas-phase ion species: 9Z=exp⁡-0.0347(ln⁡(mi))2-0.0376ln⁡(mi)+1.4662, where mi is the ion mass in amu, and Z is the ion mobility in cm2V-1s-1. Using this relationship, the measured ion mobilities map to masses as shown in Fig. b. The nonlinear mapping pushes the derived masses towards the corner with smaller values. This effect is also apparent in the derived probability density of the mass, obtained by assuming a uniform probability for the ion mobility over the trapezoid.

To demonstrate the range of plausible charging probabilities, we can now take the area of the charger ion mobilities (and, derivatively, masses) considered in Fig.  and put it through the LYF model. Specifically, we divide the ion mobility trapezoid into a grid of 1400 uniformly spaced points of ion mobility pairs (Zk+,Zk-), calculate the charging probability for every pair, and then pool the results together. The ranges of the resulting charging probabilities for charges p∈[-3,3] are shown in Fig. , along with the charging probability given by the Wiedensohler approximation. This charging probability range is the starting point for propagating the charging uncertainty into the size distribution estimates which is treated in the next section.

Table  shows the mean and 99th percentile relative deviations between the LYF model output range and the Wiedensohler approximation, for the charges p∈[-3,3]. We calculate the relative difference as 10|fp,W-fp,LYF||fp,W|+ϵ,withϵ=10-9, where the subscripts refer to the said models. The small ϵ is added to the denominator to stabilize the computation of the relative error for very small values of fp where tiny absolute differences could show as large relative differences, and ϵ=10-9 is chosen empirically based on sensitivity tests that show that charging probabilities smaller than 10-8 have a negligible influence on the inversion. Only the size ranges where the Wiedensohler approximation is valid are used in the calculation of the relative difference.

Evaluating the LYF model can be computationally heavy in practice. For example, our Python implementation takes approximately 3 min to compute a single prediction of the charging probabilities (computed for 60 size bins and considering 51 charge states between p=-25,…,25), which would make it unfeasible to carry out the uncertainty calculations where the model has to be evaluated at least a few tens of times per measurement. To make computing the LYF charging probability feasible for practical applications, we approximate the model by cubic interpolation based on precomputed values. The precomputations are carried out for 100 pairs of positive and negative ion mobilities and 40 particle diameters between the range 1 to 2500 nm and for each input value we store both the flux coefficients βq,±(d;Z+,Z-) (to enable calculation of the charging probability as in Eq. () with varying ion ratios) and the charging probability fp(d;Z+,Z-) assuming the ion ratio is unity (for maximum evaluation speed in that case). Using the interpolator is over 105 times faster than evaluating the LYF model directly, but obviously some interpolation error is introduced. In a test that considered 15 randomly chosen ion pairs, charges between -10,…,10, and 60 logarithmically spaced diameters between 1 and 2500 nm, the mean relative difference between the directly evaluated and the interpolated outputs, calculated as in Eq. (), was found to be 0.052 % and the 99th percentile relative error was 0.95 %.

We first consider the parametrization of the PSD, which falls in the category of prior assumptions even though it does not come in the form of a probability density. Two requirements guide the choice of parametrization: it should treat all orders of magnitude of the number concentration equally, so that no particular scale is favoured a priori, and it should enforce positivity of the number concentration. To see why the first point matters, consider a prior probability density for N of the form N∼U([0,107]), where U denotes the uniform distribution, i.e., the probability density is constant over the interval [0,107], which at first sight treats all values equally. However, this implicitly places a strong prior towards the highest concentrations: the probability that N is in the range [106,107] is 0.9, while for example the probability of N being in [102,103] is only 9×10-5. Ideally, we would like to be “prior ignorant” about the number concentration. The second point is obvious; number concentrations cannot be negative. There are many ways to enforce the non-negativity, but a natural choice is to build it into the parametrization itself.

The logarithmic transform satisfies both requirements and therefore we choose to represent the size distribution on a log⁡10-scale, so that x:=log⁡10(N). In addition to enforcing positivity, this has the benefit of reflecting uncertainty about the scale rather than the absolute concentration. If we place on x a uniform prior x∼U([0,7]), mimicking the idea above for N, then the probability that x is in any interval [k,k+1],k=0,…,6 is equal, which means PN∈[100,101]=…=PN∈[106,107], i.e., every decade has the same prior probability. With this parametrization, the forward operator, Eq. (), is written as 12h(x)=A10x.

In addition to positivity and scale invariance, we want to include the physical expectation that PSDs vary smoothly with particle size. To this end, we adopt a Gaussian prior with a covariance structure that penalizes sharp changes between adjacent size bins, see for example . Specifically, the covariance structure we use here is given by the squared exponential kernel where bins close in size are strongly correlated and bins far apart are weakly correlated. The covariance structure is parametrized by a correlation length l and standard deviation σd, naturally defined in log⁡10 space: 13Γpr(i,j)=σd2exp⁡-12log⁡10(di)-log⁡10(dj)2b2, where b=l2ln⁡(100). With this definition, the correlation between two sizes log⁡10(di) and log⁡10(dj) drops to 1 % when they are distance l apart. If we denote a matrix square root of the inverse of the covariance by Lpr so that LprTLpr=Γpr-1, the full expression for the prior on x is given by 14π(x)∝exp⁡-12Lprx*-x2, where x* is the prior mean.

To ensure that the prior mean does not implicitly depend on the size bin discretization, we define it in terms of the concentration density dN/dlog⁡dp rather than the bin count N. Since Ni=(dN/dlog⁡dp)i⋅Δlog⁡dp and x=log⁡10(N), the prior mean for bin i is 15x*,i=log⁡10(n*)+log⁡10(Δlog⁡dp), where n* is the assumed prior mean concentration density, which is the same for all sizes. This way, refining or coarsening the size grid shifts all x*,i by the same constant without changing the underlying physical assumption.

The exact values of σd, l, and x* are not crucial for the inversion results, in part due to the scale-agnostic parametrization. It is typically enough that they are reasonable, and in this paper we use n*=1 cm⁻³, σd=1.5, and l=0.5, meaning that unless there is evidence to the contrary (i.e., measurements), we expect very low concentrations but give this a relatively large uncertainty. In terms of the concentration density, the 95 % prior credible interval spans approximately [10-3,103] cm⁻³, covering essentially all physically plausible values. The correlation length l=0.5 means that the prior correlation between two size bins drops to 1 % over half a decade in diameter. These values can be adjusted depending on the type of measurement and expected number concentrations.

The likelihood function describes the probability of observing the measured CPC counts y given the unknown size distribution parameters. We construct it based on the forward model Eq. (), which provides the expected noiseless counts, and on considering the measurement noise.

Since the raw particle count given by the CPC is obtained by counting individual particles that are randomly distributed in the flow stream, we can assume that it follows the Poisson distribution: 16yi∼Poissonỹi,ỹi=h(x)i, For computational convenience, we approximate the Poisson noise as additive and Gaussian with Var(yi)=E[yi]=ỹi. This approximation is accurate for large counts but adds some modeling error if the number of counts is very small (<10). The observation model thus becomes 17y=h(x)+e,e∼N(0,Γe). Note that in principle the covariance should use the expected counts ỹ, but since these depend on the unknown size distribution parameters, we adopt the common approximation of using the observed counts.

In addition to Poisson counting noise, CPC measurements are affected by other uncertainties. These include (i) errors that scale with particle concentration, such as fluctuations caused by unstable flow rates or inhomogeneous air masses, and (ii) background noise from sources like leakage or homogeneous nucleation of the working fluid . The former becomes dominant at high concentrations, while the latter is independent of particle counts. We account for these additional sources of error by adding terms to the noise covariance as 18Γe=diag(y)+α2diag(y)2+β2I, where α is the noise level for concentration-dependent uncertainty, β is the noise level of the background, and I is the identity matrix. The likelihood can now be written as 19π(y|x)∝exp⁡-12Ley-h(x)2, where LeTLe=Γe-1.

The posterior density Eq. () can now be used to compute point and interval estimates of the PSD. For computational efficiency, we approximate the posterior as a multivariate Gaussian using the Laplace approximation, which is obtained by finding the maximum a posteriori (MAP) estimate and fitting a normal distribution to the local curvature of the negative log-posterior . Specifically, the MAP estimate is defined as 20xMAP=arg⁡max⁡xπ(x|y)=arg⁡min⁡x-log⁡π(y|x)-log⁡π(x)=arg⁡min⁡x12Ley-h(x)2+12Lprx*-x2 where π(y|x) and π(x) are the likelihood and the prior, respectively. To construct the Laplace approximation, we approximate the log-posterior l(x)=log⁡π(x|y) by a second-order Taylor expansion: 21l(x)≈l(xMAP)-12(x-xMAP)TH(x-xMAP), where H=-∇2l(xMAP) is the negative Hessian of the log-posterior at the MAP. Then the Laplace approximation is 22π(x|y)≈N(xMAP,H-1). In many cases the Gaussian is a good approximation of the posterior, but for instance nonlinearities such as the 10x term in the likelihood can introduce skewness or heavy tails, making the true posterior (possibly significantly) non-Gaussian. We evaluate the quality of this approximation in Appendix .

Point and interval summaries for the primary quantity of interest N are obtained by transforming from x via N=10x. Under the Gaussian approximation in x, the induced marginal posterior for each component Ni is lognormal. In particular, since the transformation is monotone, the marginal posterior median satisfies 23N^=median(N∣y)=10median(x∣y), and because each marginal xi is normal with center xMAP,i, we have median(xi∣y)=xMAP,i and thus N^=10xMAP. In this work, we report the marginal posterior median of N as the point estimate, because it is robust to possible upper-tail inflation that can arise from the Gaussian approximation and be further magnified by the nonlinear mapping N=10x (see Appendix ). We compute marginal credible intervals (highest density intervals) on the linear scale because shortest-length intervals are not invariant under nonlinear reparameterizations, and hence one computed in x and mapped to N generally differs from one computed directly in N. For the lognormal distribution implied by the Laplace approximation, the highest density interval endpoints satisfy xL,i=ci-δi and xU,i=ci+δi, where ci=xMAP,i-(ln⁡10)Var(xi∣y) with Var(xi∣y)=(H-1)ii, and δi is obtained by solving a one-dimensional equation enforcing the desired posterior mass.

In MPSS inversion, like in most inverse problems, the forward model depends not only on the primary unknowns x but also on parameters ν, such as the ion properties entering the charging probability, whose values are not precisely known. These so-called nuisance parameters are not of primary interest but cannot be ignored because they affect the estimates of x. A common approach to dealing with nuisance parameters is to assign them some fixed, plausible values and carry out the inversion as if they were known exactly. However, this simplification introduces model error which may bias the estimates of the primary unknowns and lead to underestimated uncertainties .

In the Bayesian approach, nuisance parameters can be treated as random variables with their own prior π(ν) and integrated out of the joint posterior. This process, known as marginalization, takes the uncertainty in the nuisance parameters into account automatically and propagates it into the posterior distribution of the primary unknowns. In a standard Bayesian formulation, the full posterior (the reason for the distinction “full” will become apparent below) that considers both x and ν unknown is 24πfull(x,ν|y)∝π(y|x,ν)π(x)π(ν), where we assume independent π(x) and π(ν). The marginal posterior is obtained by integrating ν out: 25πfull(x|y)∝π(x)∫Rπ(y|x,ν)π(ν)dν.

To treat the uncertainty of the MPSS charging probability in this way, we adopt the LYF model and consider its input parameters, the positive and negative ion mobilities, as nuisance parameters so that ν=(Z+,Z-). For π(ν), let R⊂R2 denote the shaded polygonal region shown in Fig. a, which represents the range of various measured ion mobilities found in the literature. Since they seem to be spread relatively uniformly over R, we model, for lack of better information, the prior as 26π(ν)=1|R|,if ν∈R,0,otherwise. where |R| is the area of R. In other words, the prior probability is uniform for all mobilities within the chosen area and zero everywhere else.

Marginalizing ν in the above full probability model can, however, lead to an inconsistency with our modelling intent for how the uncertainty in ν should propagate into the marginal posterior uncertainty of x. The reason is that in the full probability model the information from y can update both unknowns, while x and ν are multiplicatively coupled in the forward model (Eq. ) and hence not jointly identifiable from y alone: multiple pairs (x,ν) can yield indistinguishable predictions for the counts. Let X⊂RM denote the support of x. Formally, the update of ν proceeds through its marginal posterior 27πfull(ν∣y)∝π(ν)m(y∣ν),m(y∣ν):=∫Xπ(y∣x,ν)π(x)dx, i.e., through the integrated likelihood m(y∣ν) obtained by integrating out x under its prior. Because y contains essentially no direct information about ion mobilities in this application, the integrated likelihood m(y∣ν) is not driven by genuine information about ν. Instead, it is shaped primarily by how well each value of ν allows x to remain close to high-probability regions of π(x), while still explaining the data. As a result, the model in Eq. () may account for only a small range of π(ν), leading to an artificially narrow (overconfident) marginal posterior for x. This phenomenon was confirmed in our tests with the full probability model, where most of the posterior mass was assigned to a narrow subset of R when marginalizing ν.

To prevent ν from concentrating into a narrow region, we need to ensure that the data y do not update the distribution of the ν, while still allowing uncertainty in ν to propagate into the estimation of x. We can do this by adopting a modular formulation where the problematic feedback from y to ν is cut intentionally . To derive this cut posterior in our case, note first that the full joint posterior (Eq. ) can be factored into the conditional posteriors of x and ν 28πfull(x,ν|y)=π(x|ν,y)π(ν|y). In this form it is easy to see that the last term is responsible for the feedback from the data y to the nuisance parameters ν, and cutting this feedback means assuming π(ν|y)=π(ν). This forms the cut posterior 29πcut(x,ν|y):=π(x|ν,y)π(ν), and the cut marginal of x is then given by 30πcut(x|y)=∫Rπ(x|ν,y)π(ν)dν. The above equation shows that, in the cut posterior, the conditional posteriors π(x|ν,y) are averaged over π(ν), the prior distribution of the nuisance parameters. Further, we can see that the cut marginal posterior of ν is equal to its prior: 31πcut(ν|y)=∫Xπcut(x,ν|y)dx=π(ν)∫Xπ(x|ν,y)dx=π(ν), and hence in this case the data y do not update it, which is exactly what we want.

In our implementation, using the cut posterior instead of the full one does not make the problem any harder to solve, and in fact carrying out the marginalization becomes slightly simpler. To evaluate the integral in Eq. () in a numerically efficient way, we compute it with a deterministic quadrature approximation, where the prior π(ν) is divided into a regular grid {νj}j=1J⊂R, and at each νj we compute a Laplace approximation to the conditional posterior, i.e., 32π(x∣νj,y)∝π(y∣x,νj)π(x)≈N(xMAP(νj),Σ(νj)), similar to Sect. . In Eq. (), we can use ∝ for the conditional π(x∣νj,y) because νj is fixed, but this proportionality would be invalid at the joint level (Eq. ) because the normalizing constant depends on ν. The conditional posteriors are then combined to form a Gaussian mixture density 33πcut(x∣y)≈∑j=1JwjN(xMAP(νj),Σ(νj)),wj=|Cj∩R||R|,∑j=1Jwj=1, where the mixture weights wj are the ratio of the area of Cj, the rectangular cell of the grid around νj, to the full area |R|. In practice, nearly all of the weights are equal to one another and only some νj that are near the diagonal in R, where the rectangular grid cell Cj is not fully within R, have a lower weight. This corresponds to averaging each conditional posterior over the prior of ν, which is flat in our case. If we were to carry out the marginalization of the full posterior in the above way, computation of the weights would be more involved since they would be proportional to π(νj)m(y|νj). The above approach to evaluating Eq. () can also be seen as a kind of multiple imputation , although such an approach is typically stochastic.

Finally, we can draw samples from the Gaussian mixture to compute the point and interval summaries of the marginalized posterior. Similarly to Sect. , we report as the point estimate the posterior median and the credible intervals are highest density intervals calculated in the N-space.

This subsection summarizes several practical aspects of the inversion as implemented in the MPSS-UQ software package.

To account for contributions from multiply charged particles, the upper limit of the modelled size range is by default set to 2500 nm, regardless of the nominal upper size limit of the MPSS measurement. Figure  illustrates why this is necessary: although the largest nominal measurement channel in this example is 700 nm, due to multiple charging particles with diameters all the way up to the limit of 2500 nm can contribute to the channel. Truncating the modelled PSD at the nominal measurement limit (corresponding to the white dotted line in Fig. ) would ignore these contributions and lead to biased estimates. In principle, sizes above the chosen maximum of 2500 nm can also contribute, but their ambient concentration is typically low, and moreover particles larger than this are usually removed by inlet impactors.

The lower end of the modelled size range is also chosen automatically in MPSS-UQ: it extends to approximately 30 % below the smallest nominal measured diameter, with a hard limit at 1 nm. Extending the range downward helps suppress artificial edge effects and stabilizes the solution. The 1 nm limit is below the practical limit of most MPSS instruments. When reporting results, MPSS-UQ by default displays only the size interval that was actually measured.

The grid {νj}j=1J of ion mobilities used for marginalization is constructed so that the spacing in both the Z+ and Z- directions is as uniform as possible, while dividing the region R into equal segments along each axis. The default grid spacing in MPSS-UQ is 0.035 cm2V-1s-1, leading to 398 grid points. We verified the stability of posterior summaries at this discretization level with grid refinement, but denser grids can be used if required, with the usual accuracy-runtime trade-off.

In MPSS-UQ, each non-marginalized posterior is stored as the mean and covariance of its Laplace approximation, whereas the marginalized posteriors are stored as the samples drawn from the corresponding mixture distribution. For a modest number of posterior samples (thousands of draws), this requires less disk space than storing the full covariance matrices of each mixture component, while still being sufficiently accurate for posterior summaries. Further, storing the result as posterior samples retains the possibility to calculate posterior quantities, such as credible intervals or derived metrics, at different probability levels without having to re-run the inversion.

For datasets consisting of many independent measurements (e.g. time series), MPSS-UQ can distribute the inversions across available CPU cores. This parallel execution does not alter the statistical model or the posterior computation but can significantly reduce total computation time for large datasets.

To test the inversion on realistic simulated cases, MPSS-UQ implements the model aerosol distributions from Table 8.3 of p. 343. These distributions are given as the sum of three lognormal modes, and represent a wide variety of aerosol distributions, ranging from the nearly particle-free “Polar” scenario to the high particle concentration “Urban” scenario. In addition, we include manually constructed size distributions that cannot be represented as a sum of lognormal modes, such as distributions with asymmetric or irregular shapes.

The simulated measurements are generated with the following general procedure. For all simulations, the ambient temperature and pressure are set to 293.15 K and 101 325 Pa, respectively, and the number of multiple charges modelled is 10. We represent the true PSD over the range [1,2500] nm and discretize it into 500 logarithmically spaced bins, while the DMPS model is set to output 30 measurements at nominal diameters between 10 and 800 nm, also spaced logarithmically.

We first generate the noiseless expected counts ỹ=h(x) using Eq. (). To model the errors that scale with particle concentration, we perturb the expected counts as ỹ′=ỹ∘(1+αξ), where ξ∼N(0,I) and ∘ denotes elementwise multiplication. We then simulate the discrete counting process of the CPC. The total expected count rate includes contributions from both the classified aerosol sample and a constant background rate λb=β2, representing false counts from, e.g., homogeneous nucleation or detector noise. Since all individual detection events are independent, the total count follows a Poisson distribution: 34y=Poissonmax⁡ỹ′,0+λb. The background rate λb=β2 is chosen so that its Poisson variance matches the background variance β2 in the observation model Eq. (). Specifically, here we set α=0.01 and β2=0.1. Note that for computational convenience the observation model used in the inversion approximates the Poisson counting noise as Gaussian and evaluates the noise statistics at the observed counts y instead of at the true expected counts ỹ.

When inverting synthetic measurements, we must take care to avoid the inverse crime, i.e. the artificial improvement in inversion performance that occurs when the same forward model and discretization are used both to generate and to invert the data . To prevent this, we lower the discretization used in the inversion to 100 logarithmically spaced bins, over the size range that is chosen automatically as described in Sect. . Note that the inversion is not very sensitive to the precise discretization as long as it is sufficiently dense. This can be verified by increasing the number of bins until the result does not change anymore. Further, during inversion we reduce the maximum modeled number of multiple charges from 10 to 8. These changes to the forward model, and approximating the Poisson + Gaussian additive noise as Gaussian in the likelihood, ensure that the inversion model is numerically and statistically different from the data-generation model and the inverse crime is avoided.

As a preliminary verification, we applied MPSS-UQ to raw count data from a commercial SMPS system (TSI Inc. electronic classifier model 3082 coupled with TSI 3776 ultrafine CPC) measuring an atmospheric simulation chamber, using the Wiedensohler charging probability approximation. The posterior median PSDs recovered by MPSS-UQ agree with the standard TSI inversion in both shape and magnitude (not shown), confirming that the proposed method produces results consistent with an established reference when the same charging model is used. Despite the fact that the two approaches likely differ in their implementation of the SMPS forward model, the majority of the TSI point estimates fall within the MPSS-UQ 95 % credible intervals.

We now apply the MPSS-UQ inversion to field data from a month-long DMPS deployment in December 2024 at the Puijo measurement station in Kuopio, Finland, where the charging probability is no longer assumed known and the full marginalization is used. The DMPS consists of a Nickel-63 charger, a Vienna-type cylindrical DMA operated with a closed-loop sheath flow, and an Airmodus A20 CPC. The instrument configuration and parameters are summarized in Table . The DMPS is set to measure 29 logarithmically spaced channels between nominal diameters of 11.6 and 800 nm, and in the inversion we use 70 size bins. The whole size range is measured every 6 min resulting in approximately 7400 scans. Temperature and pressure were recorded for each scan, and they are used to update the DMPS instrument matrix during inversion at every scan.

The full marginalization of the entire dataset with MPSS-UQ required about 10 min on a standard laptop (Intel(R) Core(TM) i7-12700H processor, 32 GB RAM). Figure  displays the posterior median PSD (panel a), the associated relative uncertainty (panel b), and the total particle number time series (panel c). The white vertical gaps correspond to instrument maintenance periods. We quantify the uncertainty as the relative width of the 95 % highest density credible interval, 35W=CI95,high-CI95,lowN^⋅100%, where CI95=[CI95,low,CI95,high] denotes the 95 % credible interval and N^ the posterior median. We report relative rather than absolute uncertainty because MPSS measurements follow a Poisson counting process whose standard deviation grows with the expected count as ỹ. As a consequence, the absolute posterior uncertainties are roughly proportional to concentration, so that an absolute uncertainty plot is visually similar to the median estimate itself and therefore conveys little additional information. The relative width W removes this scaling: a value of W=100 % always means that the width of the 95 % credible interval equals the posterior median, regardless of the concentration level. This makes W an interpretable and scale-independent measure of uncertainty that is consistent with both the Poisson noise model and the log⁡10-scale parametrization of the PSD. Note that the colour scale in panel (b) is divided into three regions to accommodate the wide dynamic range of W. For W≤1000 %, the color map shows full contrast in the range where the data most meaningfully constrain the PSD. Values between 1000 % <W≤10000 % are shown in grey to indicate a zone where the posterior is narrower than the prior but the estimate remains highly uncertain. Values exceeding W=10000 %, where the data provide little constraint beyond the prior, are shown in white. Finally, panel (c) shows the total particle number concentration together with its 95 % credible interval. We discuss some more derived quantities in Sect. .

Multiple new particle formation (NPF) events, as well as days with low particle concentrations, are visible in the month-long dataset. To examine differences between the marginalized and fixed-mobility inversions in detail, we analyze two representative days: an NPF event on 10 December 2024 and a low-concentration day on 25 December 2024.

Figure  shows the inversion results for the NPF event day. The posterior median PSDs in panels (a) and (b) differ only slightly between the two approaches, but this is expected: the posterior median of the marginalized inversion corresponds approximately to using an “effective” set of charging probabilities averaged over the mobility prior region R, and the fixed mobilities used in the non-marginalized inversion lie near the center of this region. The relative credible interval widths W (shown in panels c and d), however, are considerably larger in the marginalized case than in the non-marginalized one. The ratio of W between the marginalized and non-marginalized inversions ranges from 1 to 9 across the measured size range. Further, the non-marginalized intervals suggest there is a band from roughly 20 to 70 nm, corresponding to bins with the highest concentration of particles, that has much less uncertainty than the surrounding sizes. The marginalized intervals do not support this, however, and instead they indicate that the uncertainty is roughly constant between 20 and 250 nm, increasing only slightly towards smaller sizes. As with the synthetic data examples earlier, the differences are most apparent in total particle numbers Ntot (panels e and f). While the posterior medians of Ntot are similar, the ratio of marginalized to fixed 95 % credible interval widths varies between 4 and 18.

Figure  shows the results for the low-concentration day. The qualitative differences between the two inversion approaches remain the same, but the ratios are somewhat smaller: the W ratio varies between 1 and 4 across the size range, and the corresponding Ntot credible interval width ratio varies between 3 and 8, compared with 1–9 and 4–18 for the NPF day, respectively. This is consistent with a general pattern across the full dataset, where the ratio increases with total particle number concentration (correlation r=0.78). In other words, the degree to which the fixed-mobility inversion underestimates the uncertainty grows with concentration.

To illustrate inversion output at the level of a single measurement, Fig.  presents the posterior median and 95 % credible intervals for one scan from the NPF event. The difference between the marginalized and fixed-mobility credible intervals is clearly visible, consistent with the patterns observed in the full-day results above.

To interpret the uncertainty plots, it is helpful to understand the main factors that determine the width of the posterior credible intervals.

The factor that has the most straightforward effect is the particle concentration level. Because particle counting follows Poisson statistics, relative counting noise decreases as the count rate increases. Higher concentrations therefore provide more informative measurements and lead to tighter posteriors than do low concentrations. This is visible in the relative uncertainty plots, where the smallest concentrations typically correspond to the highest values of W.

Several physical effects reduce inversion accuracy at the edges of the size range. Charging efficiency drops rapidly below roughly 20 nm, the proportion of multiply charged particles becomes more significant at larger diameters, and sub-20 nm particles additionally suffer stronger diffusion losses that reduce the number that reach the CPC. Together, these effects make the smallest and largest bins more weakly constrained and lead to wider credible intervals in those regions.

A factor particularly relevant to the present work is the assumed range of ion mobilities. The width of the mobility prior region R determines how much charging-related uncertainty enters the inference. A narrower R, i.e., better prior knowledge of ion mobilities (for example through tighter control of carrier gas composition, or even directly measuring the ion properties) results in correspondingly narrower posterior credible intervals. In the limiting case where ion mobility is treated as known, the marginalised posterior collapses to the non-marginalised one, which reproduces the narrow, but typically overconfident, intervals shown in Sect. .

The size resolution and number of measurement channels also play a role. The amount of independent information available from the data depends on the number, spacing, and width of the channels. Instruments with more channels per scan, such as a typical SMPS compared to a typical DMPS, may support finer structure in the inferred PSD, provided the prior correlation length is adjusted accordingly. For example, in the SMPS verification we used a correlation length of l=0.25 decades while still obtaining smooth, realistic results, whereas the sparser DMPS data required l=0.5 decades to avoid fitting noise.

Let us now discuss some limitations of the present work.

First, the inversion presented here treats each measurement scan independently, without modelling temporal correlations between successive scans. In reality, the PSD evolves continuously, and consecutive scans are not independent. A temporal model, for example based on Kalman filtering or smoothing , could exploit this continuity to further constrain and smooth the posterior estimates. Extending the current framework to include temporal dependence is a natural direction for future work.

The present work focuses on charging probability as the dominant source of model uncertainty, but other factors also contribute, including uncertainties in the DMA transfer function, CPC counting efficiency, flow rates, and particle losses in the sampling lines. These could in principle be incorporated into the Bayesian framework as additional nuisance parameters to be marginalized, though the computational cost increases with each added parameter and may quickly become prohibitive. The effect of additional sources of uncertainty could potentially be taken into account in an approximate manner using the Bayesian approximation error approach , which accounts for model uncertainties by statistically characterizing their effect on the observations and including them into the likelihood. This would allow additional sources of uncertainty to be included at a fraction of the computational cost of full marginalization. A systematic comparison of the relative contributions of the sources of uncertainty (cf. ) would help prioritize which ones merit explicit treatment and which can be treated, e.g., with approximate methods.

As discussed in Sect. , there are several models available for describing bipolar charging kinetics , and all results presented in this work are conditioned on the validity of the chosen LYF model . The Bayesian framework itself, however, is not tied to this particular model and any charging model that maps ion properties to charge fractions can be substituted without changes to the inversion methodology. The LYF model, like all charging models, includes approximations and uncertainties in its description of the charging process, such as the assumption that the particles are spherical or that each ion-particle collision results in a charge transfer . The model also requires input parameters other than the ion mobilities (and masses), such as the relative permittivities of the particles and ions (χp=80 and χi=5 in this work), which we have taken as fixed here but which can also introduce additional bias, especially for particles composed of low-permittivity materials . We have also assumed that the concentrations of positive and negative charger ions are equal, although a charge imbalance can arise from the higher wall-loss rate of the more mobile negative ions in the plumbing downstream of the charger . Marginalization over this parameter is implemented in MPSS-UQ but was not explored in this study.

Finally, in the treatment of charging uncertainty in this work we approximated the ion properties as single values of positive and negative mobility, whereas in reality the charging environment contains a mixture of ion species with different masses, mobilities, and structures. Both ambient mass spectrometric measurements and laboratory tandem mobility–mass analyses of radioactive source ions have shown that the ion population is chemically diverse, temporally variable, and system-dependent. The standard Fuchs charging equations can be extended to account for such distributions of bipolar ion mobilities , and the resulting charging ratios can differ from those computed using a single ion type. In the context of the present work, this means that the true charging-related uncertainty may be broader than what our two-parameter marginalization over (Z+,Z-) captures, since it does not account for the polydispersity of the ion population itself. A more realistic representation of the ion mobility distribution in the marginalization is a natural extension of this work, though it would require reliable characterization of the ion population under the specific charger and gas conditions used, which remains a considerable experimental challenge.