the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Approximate sequential Bayesian filtering to estimate ^{222}Rn emanation from ^{226}Ra sources using spectral time series

### Stefan Röttger

### Annette Röttger

A new approach to assess the emanation of ^{222}Rn from
^{226}Ra sources based on *γ*-ray spectrometric measurements is
presented. While previous methods have resorted to steady-state treatment of
the system, the method presented incorporates well-known radioactive decay
kinetics into the inference procedure through the formulation of a
theoretically motivated system model. The validity of the ^{222}Rn
emanation estimate is thereby extended to regimes of changing source
behavior, potentially enabling the development of source surveillance
systems in the future. The inference algorithms are based on approximate
recursive Bayesian estimation in a switching linear dynamical system,
allowing regimes of changing emanation to be identified from the spectral
time series while providing reasonable filtering and smoothing performance
in steady-state regimes. The derived method is applied to an empirical
*γ*-ray spectrometric time series obtained over 85 d and is able to
provide a time series of emanation estimates consistent with the physics of
the emanation process.

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^{222}Rn emanation from

^{226}Ra sources using spectral time series, J. Sens. Sens. Syst., 12, 147–161, https://doi.org/10.5194/jsss-12-147-2023, 2023.

^{222}Rn is an odorless, colorless, radioactive noble gas, occurring
naturally in the environment as part of the primordial ^{238}U decay
chain. Due to its high mobility in the environment, radon can accumulate
inside buildings where, in conjunction with its decay products (short-lived
progeny; SLP), it is responsible for the most significant natural exposure
of the general public to ionizing radiation, which is an important
consideration for incidents of lung cancer (Darby et al., 2005). In the
2013/59/EURATOM treaty, 300 Bq m^{−3} was stipulated as the
action level for indoor ^{222}Rn activity concentrations, beyond which
mitigation is required. On the other hand, however, measurements of outdoor
^{222}Rn activity concentration at environmental levels have numerous
beneficial applications in environmental sciences, including (but not
limited to): as a tracer of terrestrial influence on air masses (Chambers et
al., 2016, 2018); as a tool for classifying the atmospheric mixing state
(e.g., Perrino et al., 2001; Williams et al., 2013, 2016; Chambers et
al., 2019b, a); and as a tool for estimating integrated local- to
regional-scale emissions of trace gases with similarly distributed sources,
such as CH_{4}, N_{2}O, or CO_{2} (Levin, 1987; Biraud et al., 2002;
Laan et al., 2010; Levin et al., 2021). For these reasons, there is an
interdisciplinary need for SI-traceable calibration procedures at low
activity concentrations for atmospheric radon monitors, and the
associated realization and dissemination of the unit Bq m^{−3},
which can only be feasibly realized through emanation sources rather than
gaseous standards (Mertes et al., 2020; Linzmaier and Röttger, 2013;
Röttger et al., 2014). In these prior studies, a method was presented
that enabled *γ*-ray spectrometric data from an open ^{226}Ra source
(i.e. emitting ^{222}Rn) to be used to estimate the resulting activity
concentration of ^{222}Rn in a closed volume, by measuring the activity of
residual SLP in the source to quantify the ^{222}Rn emanation. In recent
years, there has been a trend towards the use of dynamic calibration
conditions for ^{222}Rn (e.g., Fialova et al., 2020), eliminating the need
for the long build-up period associated with static conditions. However, it
has been demonstrated that emanation from most materials demonstrates a
strong dependence on humidity and temperature as a result of changes in
diffusion properties (Janik et al., 2015; Stranden et al., 1984; Zhou et
al., 2020), which generally results in a correlation between emanation and
environmental parameters. In these cases, previously established methods to
determine the amount of emanating ^{222}Rn fail over a considerable
time span, since the dynamical processes taking place in the source are not
accounted for. Hence, experimental investigations of source behavior under
different environmental conditions are hardly possible using established
methods. Yet, this capability is particularly important for the realization
of in situ field calibrations of large volume atmospheric ^{222}Rn
monitors, since dynamic methods would simplify the technical aspects of in
situ field calibrations considerably. Said limitations are pointed out and
discussed in the theoretical section of this work and have not been stated
nor addressed elsewhere. A possible way to correct for such environmental
influences, however, lies in determining the amount of emanating ^{222}Rn
in near real time. We present herein that this can be achieved through
continuous measurement of spectrometric time series of the ^{222}Rn
emanation sources and a suitable method for data analysis that addresses the
dynamic behavior of the system. The main contributions of the present work
are the discussion of the limitations of established methods and the
derivation and implementation of an alternative method, based on the
well established computational techniques of recursive Bayesian estimation.
First results obtained by application of the proposed method to experimental
data are presented, which illustrate the theoretically motivated limitations
and which can be well explained on the basis of the physical processes
taking place in the emanation source, justifying the correctness and
superiority of the presented method.

## 2.1 Radioactive decay kinetics

Radioactive decay chain kinetics are described by a linear time invariant
(LTI) system of ordinary differential equations. Historically, this has been
expressed in terms of the Bateman equations (Bateman, 1910), but recently
this has been more conveniently written in matrix form (Pressyanov, 2002;
Levy, 2019; Amaku et al., 2010). Undistorted radioactive decay kinetics of a
decay chain of *n* nuclides without, or with only negligible, branching, as
in the case of the ^{226}Ra decay chain, can thus be written concisely as
Eq. (1). The fundamental matrix **K** can be constructed such that it
consists of the respective decay constants *λ*_{i} on its diagonal
and generally on its first superdiagonal, while all other entries are 0.
** A** denotes a vector consisting of the activities of the
respective nuclides in the decay chain. Equation (1) can readily be discretized
using the matrix exponential of

**K**, which is in a form that can be conveniently computed by diagonalization of

**K**, e.g., through its (symbolically accessible) Jordan canonical form or its Eigen decomposition.

## 2.2 ^{222}Rn emanation

The release of ^{222}Rn from a ^{226}Ra source distorts the dynamics
described above, due to the introduction of an additional sink-term *η*
(^{222}Rn atoms released per unit time), which is not directly
quantifiable experimentally. Previously, a method was presented to measure a
steady-state emanation coefficient, previously understood to be the ratio of
exhaled and generated ^{222}Rn atoms (Mertes et al., 2020; Linzmaier and
Röttger, 2013) at any instant in time.

where *χ* is the emanation coefficient, *η* is the release of
^{222}Rn atoms per unit time, and ${A}_{\text{Ra-226}}^{\mathrm{S}}$ is the
^{226}Ra activity of the source.

By first principles, however, the ^{222}Rn activity
${A}_{\text{Rn-222}}^{\mathrm{S}}$ must follow first-order continuity as in
Eq. (3).

where ${A}_{\text{Rn-222}}^{\mathrm{S}}$ and ${A}_{\mathrm{Ra}-\mathrm{226}}^{\mathrm{S}}$ are the activities of the respective nuclides in the source material.

To this point, the best method to derive *χ* has been by measuring the
ratio of SLP and ^{226}Ra activities within the source, most commonly by
*γ*-ray spectrometry, where *χ* is defined as

According to Eqs. (2) and (3) however, this is only applicable under
steady-state conditions. This limitation was not acknowledged or discussed
in earlier work. Consequently, application of this simple method is
restricted either to regimes of a completely stabilized source, or a closed
accumulation volume for the emitted ^{222}Rn. In closed volumes, the
errors associated with neglecting the dynamics in the source are negligible,
since the volumetric ^{222}Rn follows the same dynamics. As such, after
initial equilibration, Eq. (4) holds, even with changing *χ*. However, in
this case *χ* is more accurately understood to be a partitioning
coefficient of ^{222}Rn between the source and the closed volume, rather
than the definition given in Eq. (2). The result of Eq. (3) is that the
measurable ${A}_{\text{Rn-222}}^{\mathrm{S}}\left(t\right)$ is given by
the convolution of the latent *η*(t) with an impulse
response function that is defined by the radioactive decay kinetics. Thus,
the estimation of *η*(t), or equivalently *χ*(t), based on measurements of ${A}_{\text{Rn-222}}^{\mathrm{S}}\left(t\right)$ is an
inverse problem and cannot be carried out feasibly by simple numerical
estimation of the gradient (cf. Eq. 3) due to the ubiquitous Poisson noise
associated with radioactivity measurements. Moreover, in dynamic
calibrations, *η*(t) must be expected to vary with changes
in the environmental conditions. It is also expected that this dependency
will be strongly related to the source design and its specific production
parameters. A further consideration is that, when using this method, it is
not readily possible to accurately measure correlations of *χ*(t) with environmental conditions on timescales smaller than at least
five half-lives of ^{222}Rn without considering the decay kinetics. These
limitations make it infeasible to use the existing direct approach to
continuously estimate ^{222}Rn release as required in dynamic calibration
procedures, or to derive correction factors for different environmental
conditions, since the time required would be far too long for such
measurements due to the half-life of ^{222}Rn of approximately 3.8 d.
Here we present and discuss a new approach that embraces the described
behavior and enables the release of ^{222}Rn from sources to be estimated
more accurately using continuous spectrometric measurements with the
generalization to non-steady-state situations. The algorithms and
assumptions presented have been chosen such that, in the future, the
necessary computations would be feasible on relatively low-power devices
(e.g., current single-board computers) in an online fashion.

## 2.3 Recursive Bayesian estimation and model formulation

Recursive Bayesian estimation describes a class of algorithms to perform
statistical inference in dynamical systems that can be modeled by a
(first-order) Markov process. The general idea of these methods is to
sequentially form priors for a state vector ** x** and a
dynamical model of the system, and use noisy measurements

**related to**

*y***to correct them through a measurement model and Bayes theorem (Särkkä, 2013). This method can be used to perform statistical inversion, in that it enables the estimation of latent terms whose values in a specific dynamical system are only partly measured. This is closely related to the case described in the previous section, given that the state vector, the dynamics, and the measurements can all be suitably modeled. In this setting, two collections of conditional probability distributions are of interest, which are commonly called the filtering distributions $p\left({\mathit{x}}_{n}|{\mathit{y}}_{\mathrm{1}:n}\right)$ and the smoothing distributions $p\left({\mathit{x}}_{n}|{\mathit{y}}_{\mathrm{1}:N}\right)$, where the notations 1:**

*x**n*and 1:

*N*denote the collection of all data observed up to time

*t*

_{n}and the collection of all data, respectively, and

*t*specifies an instant in time in the set of measurement times,

*T*, indexed by

*n*. In cases where

**follows a first-order Markov process, and under certain conditional independence assumptions, the definition of $p\left({\mathit{x}}_{n}|{\mathit{y}}_{\mathrm{1}:n}\right)$ and $p\left({\mathit{x}}_{n}|{\mathit{y}}_{\mathrm{1}:N}\right)$ can be expressed recursively (Särkkä, 2013; Särkkä and Solin, 2019). Prediction of the state vector**

*x*

*x*_{n}at time

*t*

_{n}given a collection of measurements ${\mathit{y}}_{\mathrm{1}:n-\mathrm{1}}$ up to time

*t*

_{n−1}and the filtering density of the state at time

*t*

_{n−1}, $p\left({\mathit{x}}_{n-\mathrm{1}}|{\mathit{y}}_{\mathrm{1}:n-\mathrm{1}}\right)$ is given by the Chapman–Kolmogorov Eq. (5) (Särkkä and Solin, 2019).

Upon observation of *y*_{n}, the density predicted by Eq. (5) is
corrected into a filtered posterior using Bayes theorem with the measurement
likelihood *p*(*y*_{n}|*x*_{n}), assumed to be conditionally independent of the past
states and measurements, i.e. $p\left({\mathit{y}}_{n}|{\mathit{x}}_{\mathrm{1}:n},{\mathit{y}}_{\mathrm{1}:n-\mathrm{1}}\right)=p\left({\mathit{y}}_{n}|{\mathit{x}}_{n}\right)$
(Särkkä, 2013).

Conversely, smoothing refers to computing the density of the state given all measurements in a specific time interval, or the complete collection. In most cases, smoothing can be defined recursively using information inferred from the filtering and starting a backward recursion at the last time instant at which the smoothing and filtering densities are equal. Formally, the smoothing density is recursively defined by Eq. (6) for the above conditional independence assumptions.

The most notable examples of these types of algorithms are the Kalman filter
(Kalman, 1960) and the Rauch–Tung–Striebel smoother (Rauch et al., 1965),
which yield the optimal estimators for discrete systems of linear dynamics
and independent Gaussian noise, where the filtering and smoothing can be
carried out in *O*(T) time.

For the problem at hand, a stochastic differential equation (SDE) is needed
to express the (time-varying) uncertainty associated with the latent
continuous variable *η*(t). This can be seen as an
application of the latent-force models introduced in Alvarez et al. (2009), whose link with
Bayesian filtering was previously established in Särkkä et al. (2019) and
Hartikainen and Särkkä (2012). The specific choice of SDE is
subjective but allows the properties of the resultant functions to be
constrained, and can be used to encode prior knowledge (Särkkä and
Solin, 2019). In this work, without any claims of optimality, the choice was
made to use the zero-mean, mean-reverting Ornstein–Uhlenbeck process for the
first derivative of *η*, i.e., Eq. (7), resulting in constraining *η* to somewhat smooth functions of certain autocorrelation.

where d*β*_{t} describes the increments of a
scalar standard Brownian motion.

While this choice is not entirely representative of the physical mechanisms
related to *η*, in that, for example, it allows for negative values of
*η*, it results in a Gaussian process for which the inference procedure
has convenient conjugacy and thus a closed-form solution, such that the
resultant algorithms are suitable for online operation on low-power
computational hardware that can be reasonably used to monitor an emanation
source during its operation. It should also be noted that Eq. (2) was
formulated with this in mind, in the sense that *η* is modeled as
state-independent, rather than state-dependent, as would likely be more
realistic for a diffusive process like ^{222}Rn emanation. However, these
theoretical inaccuracies did not manifest in practice with the experimental
data presented in Sect. 3, given that *η* is far enough from 0 and the
collected data are strong enough, while only approximate inference is of
interest. The system is thus modeled to follow the combined SDE given in
Eqs. (8)–(11) in terms of the Itō stochastic integral (Särkkä and
Solin, 2019; Hartikainen and Särkkä, 2012).

## 2.4 Measurement model for integrating spectrometric data

Unlike most applications of such filtering algorithms for discretized LTI
systems, here the supporting measurements cannot be made at instantaneous
moments in time because disintegrations of a specific nuclide can only be
recorded over a finite time interval, *r*_{n}, indexed by *n*. In routine
radioactivity analysis, this behavior results in decay or ingrowth during
measurement corrections. However, in the present case, an additional
contribution to the uncertainty results from unknown changes of *η* over
the integration time.

The way we have chosen to model this behavior in the present study is to
start by stating that the uncorrupted (i.e., noise-free) measurements are
given by Eq. (12), where, for now, we assume that **H** is known
deterministically.

where **H** is a matrix that maps the state integral to the
measurement space.

The elements of **H** are related to the counting efficiency
of the setup and/or nuclide. Note that, in principle, it would be possible
to choose **H** to directly model some region of the
spectrum, but we chose to use derived peak areas or even spectrum integrals
as the input data such that the elements of **H** are just
the counting efficiencies. More elaborate modeling in the spectrum space was
not attempted, since the filtering algorithms generally require inversion of
the residual covariance matrix in the measurement space, and thus scale
approximately with *O*(*k*^{3}) where *k* is the dimensionality
of the measurements.

Since integration is a linear operator, the joint distribution of
*x*_{n−1}, *x*_{n}, and *y*_{n} clearly has a
Gaussian density, and hence $p\left({\mathit{x}}_{n},{\mathit{y}}_{n}|{\mathit{y}}_{\mathrm{1}:n-\mathrm{1}}\right)$ is readily found by marginalizing over
*x*_{n−1}, given that the previous time step posterior filtering
distribution $p\left({\mathit{x}}_{n-\mathrm{1}}|{\mathit{y}}_{\mathrm{1}:n-\mathrm{1}}\right)$ is
already known (and Gaussian). This joint density is derived in Appendix A1.
It is assumed that the measurement *y*_{n} is related to the state
*x*_{n} by integrating from ${t}_{n-\mathrm{1}}+{\mathit{\delta}}_{n}$ to
${t}_{n}={t}_{n-\mathrm{1}}+{\mathit{\delta}}_{n}+{r}_{n}$, such that arbitrary integration intervals
*r*_{n} and arbitrary time offsets *δ*_{n} are possible (e.g., if
measurements are skipped or delayed). In other words, each time instant
where the density of ** x** is inferred coincides with the
endpoint of each spectrum acquisition. This approach intrinsically accounts
for the ingrowth and decay during the finite integration time, but more
importantly, additionally estimates the uncertainty arising from the
possible change of

*η*within the integration time. Given that $p\left({\mathit{x}}_{n},{\mathit{y}}_{n}|{\mathit{y}}_{\mathrm{1}:n-\mathrm{1}}\right)$ under the present model is thus accessible, the posterior filtering distribution $p\left({\mathit{x}}_{n}|{\mathit{y}}_{\mathrm{1}:n}\right)$ is found by conditioning onto the observed value for

*y*_{n}using the well-known conditioning formula for multivariate Gaussians.

Radioactivity measurements generally follow Poisson statistics. In the
framework of Bayesian inference and recursive Bayesian estimation,
non-Gaussian noise considerably complicates the inference procedure, since
the measurements are then no longer a Gaussian process and thus no longer
conjugate with the state. For this reason, there is no exact closed-form
solution for the filtering recursions in the case of non-Gaussian noise.
Considerable work has been done to address this, including sampling
procedures like Markov chain Monte Carlo, particle filtering, or expectation
propagation (Minka, 2013), or by assuming that all arising probability density functions (PDFs) are Gaussian
combined with an estimation strategy for the moments (e.g., unscented
Kalman filter; Julier et al., 2000). Generally, we found such approaches
unsuitable due to their computational complexity but also because the
measurements can be well approximated as Gaussian (due to the high number of
counts that are being observed). Instead, we approximate the measurement
likelihood as Gaussian where the moments are evaluated from the previous
time step filtering distribution of ** x**. Ebeigbe et al. (2020)
proposed that the covariance matrix could be estimated from the mean of the
filtered state in a different context, which we adopted.
Combining this with the results of Appendix A1, we obtain Eq. (13)
as an approximation of the joint density $p\left({\mathbf{x}}_{n},{\mathbf{y}}_{n}\phantom{\rule{0.125em}{0ex}}\mathrm{|}\phantom{\rule{0.125em}{0ex}}{\mathbf{y}}_{\mathrm{1}:n-\mathrm{1}}\right)$.

where

and where the index *n* has been dropped on *r* and *δ* for notational
brevity and the included matrices are given as follows:
**R** is a variance term that accounts for the variance
contribution of the background count rate, which is estimated from prior
background measurements and is assumed constant over time, and *μ*_{n−1} and **Σ**_{n−1} denote the
mean and covariance matrix of the filtering distribution at the previous
time step.

## 2.5 Extension for strong variations in emanation characteristics or discontinuities

The variance *σ* of the Brownian motion process in Eq. (7) allows a
linear dynamical model as shown in Sect. 2.3 and 2.4 to be tuned in a
trade-off-like fashion for one of two properties. Predictions are good in
times of relatively constant *η* and low *σ*, such that the
measurement noise is well filtered at the expense of fidelity in response to
steep changes in *η*, or vice versa. In practical situations however,
where the humidity can change rapidly and is known to affect the emanation,
a period of re-equilibration is induced in the source, before it returns to
somewhat stable behavior. Therefore, experimental time series of the radon
source spectra typically show properties that are not well addressed by a
single such model of linear dynamics due to the described distinct regimes.
This kind of situation is strongly related to the tracking of maneuvering
targets, a field in which Bayesian recursive estimation is well established.
To obtain good filtering performance in the sense of providing relatively
smooth estimates with small uncertainty in case of constant *η* while
retaining the ability to quickly react to steep gradients with associated
larger estimation uncertainty, one approach suggested by Nadarajah et al.
(2012) and Mazor et al. (1998), among others, is to use the interacting multiple
model (IMM) recursive estimator. This is commonly used for object tracking
and in this work was adopted to refine the procedure outlined in the
previous sections. In the IMM, multiple filters operate on the data at the
same time, and their output is mixed based on the likelihood of their
measurement predictions. In this way a second, discrete, first-order
Markov process describing a discrete random variable ${s}_{{t}_{n}}$ that
indexes into the several applicable and differently parameterized linear
dynamical models is formally introduced and evolves according to some,
potentially parameterized, transition matrix **Π**. As a
result, the filtering and smoothing distributions become the compound
distributions $p\left({s}_{n},{\mathit{x}}_{n}|{\mathit{y}}_{\mathrm{1}:n}\right)$ and $p\left({s}_{n},{\mathit{x}}_{n}|{\mathit{y}}_{\mathrm{1}:N}\right)$,
respectively, which now also carry probabilistic information regarding the
active model index *s*_{n}.

These compound distributions can be decomposed into discrete and continuous
components which are approximately represented as mixtures of Gaussians.
This kind of system is also called a switching linear dynamical system
(SLDS). In the SLDS, exact filtering is not computationally feasible
(Barber, 2006; Hartikainen and Särkkä, 2012), since the filtering
distribution is a mixture of Gaussians whose components are being multiplied
by the number of models at each time step resulting in the exponential
growth of components. Most approaches for approximation, like the one
employed here, replace the resultant mixture at each step of filtering and
smoothing with a smaller one, limiting the number of kept components in the
Gaussian mixtures to some fixed upper value *I*. In case of filtering, these
types of algorithms are referred to as Gaussian sum filtering (GSF). Figure 1 gives an outline,
both in terms of a component-wise (i.e., marginalized only over the mixture
component indices *i*_{n}) and a marginal view (i.e., marginalized over both
the model indices *s*_{n} and the component indices *i*_{n}) of the applied
GSF method for the final SLDS model for the example of two Gaussian
components per mixture. The basis of the prediction and correction steps for
each combination of active model and prior components, jointly indexed by
*s*_{n}, *s*_{n−1}, and *i*_{n−1}, is detailed in Sect. 2.3 and 2.4. By the
discretizations carried out in Sect. 2.3 and 2.4, it is thus implicitly
assumed that the active model index can only switch after the observation of
each spectrum but not during the integration time of each spectrum. The
probabilities $p\left({s}_{n}|{\mathit{y}}_{\mathrm{1}:n}\right)$ are also estimated by the algorithm by evaluation
of the likelihood of measuring *y*_{n} in each component of the
prediction step density and marginalization over *x*_{n} and
associated *i*_{n}, respectively. For further details on the GSF method, the
reader is directed to Barber (2006). The results of the GSF algorithm are
the (unsummed) factors on the right hand side of the following approximation
of the decomposed filtering distribution, which consist of said Gaussian
mixtures:

where *s*_{n} is the model index at time step *t*_{n}, *i*_{n} is the
component index of the approximating Gaussian mixtures, and ${w}_{{s}_{n},{i}_{n}}$
are the associated Gaussian mixture weights.

In this setting, the smoothing distribution is also a compound distribution in which the components of each mixture are multiplied within each smoothing step in a backwards recursive formulation by the number of linear dynamical models. Therefore, smoothing is also only possible approximately, once again, on the basis of approximating each arising Gaussian mixture with a smaller one. One way to approximately obtain smoothed results in the SLDS setting is thus given by the expectation correction (EC) algorithm introduced in Barber (2006) and shown therein to provide state-of-the-art results, both in terms of computational efficiency and accuracy, using the results of the GSF forward pass and performing a backwards recursion through the time series. The EC algorithm requires, analogous to the GSF forward pass, the propagation and correction methods, in the form of the filtering and smoothing steps, for the parameterized linear dynamical system outlined in Sect. 2.3 and 2.4 acting on each Gaussian mixture component. In this case, however, the integrating behavior of the measurements does not lead to any required adjustments, and the applied equations are thus exactly analogous to the classical Rauch–Tung–Striebel smoother that represent a formal reversal of the dynamics. These have also been used in the original presentation of the EC algorithm (Algorithm 5 in Barber, 2006). Figure 2 provides a graphical illustration similar to the filtering method in Fig. 1 of the EC smoothing backward pass. The EC algorithm provided in Barber (2006) was used without further modifications, and for more information on this algorithm, the reader is directed to this work.

While not a main contribution to this paper, for completeness sake and to facilitate possible re-implementation, both the GSF forward pass and the EC backward pass are outlined in pseudo-code in Appendix A2 in the way they have been implemented here.

To model the two distinct regimes outlined before, two linear dynamical
models are used that share their *γ* values, but for one (cf. Eq. 8), *σ* is constrained to a small value. Consequently, one of the
models corresponds to regimes of changing *η* and the other to near
constant *η*. This approach allows us via the model index probabilities
$p\left({s}_{n}|{\mathit{y}}_{\mathrm{1}:n}\right)$ and
$p\left({s}_{n}|{\mathit{y}}_{\mathrm{1}:N}\right)$ obtained
from the GSF and the EC algorithms to also probabilistically identify
regimes of constant radon emanation within a time series of recorded
spectra, even when the retained activity of ^{222}Rn is still in a
re-equilibration period, as will be illustrated in the experimental results
presented later. In that sense, and by construction of the two models, the
model index probabilities can be physically interpreted as the probability
of the source to currently have stable emanation characteristics. A non-zero
*σ* acting as a regularization term is, however, still needed in the
case of the model corresponding to the constant regimes because otherwise
the approximations used within both the GSF and the EC algorithm become
numerically unstable. The final model contains four unknown parameters,
*σ*,*γ*, and the components of **Π**, which
is row-wise normalized and whose components are thus parameterized by two
parameters. These four parameters are tuned by minimizing the (approximate)
negative marginal log likelihood of the measurement series, $-{\sum}_{n}\mathrm{ln}p\left({\mathit{y}}_{n}|{\mathit{y}}_{\mathrm{1}:n-\mathrm{1}},\mathit{\sigma},\mathit{\gamma},\mathbf{\Pi}\right)$, that
is accessible in the GSF forward pass (see Algorithm 1 in Appendix A2). For
the purposes of this initial investigation, we assume the uncertainty
arising from uncertain parameters is negligible in light of the counting
statistics, the uncertainty encoded in the prior of the initial state,
*p*(*x*_{0}), and the components of **H**.

## 2.6 Propagation of the detection efficiency uncertainty

In practice, the components of **H** are not known without uncertainty,
and the uncertainty associated with **H** is the most significant
contribution to the combined uncertainty of *η* in most practical cases.
The outlined formalism above is a way to obtain estimates for $p\left({\mathit{x}}_{n}|{\mathit{y}}_{\mathrm{1}:n},\mathbf{H}\right)$ and $p\left({\mathit{x}}_{n}|{\mathit{y}}_{\mathrm{1}:N},\mathbf{H}\right)$ or the extended compound
distributions $p\left({s}_{n},{\mathit{x}}_{n}|{\mathit{y}}_{\mathrm{1}:n},\mathbf{H}\right)$ and $p\left({s}_{n},{\mathit{x}}_{n}|{\mathit{y}}_{\mathrm{1}:N},\mathbf{H}\right)$ detailed in the previous
section, respectively. For notational simplicity, the following is formulated
with respect to the single-model case but applies analogously to the SLDS
model. Formally, the inclusion of this systematic measurement uncertainty
into the filtering or smoothing result, respectively, is given by Eq. (15),
which is not tractable.

In the present case, we assume **H** to be constant with respect to
time and that its distribution is known from previous measurements,
independent of the collection *y*_{1:N}. Generally, the uncertainty
in **H** associated with, for example, a prior calibration
procedure, can reasonably be described by this approximation, if
$\frac{\mathrm{d}\mathbf{H}}{\mathrm{d}t}=\mathrm{0}$. This allows rewriting Eq. (15)
to obtain Eq. (16); however, the result remains computationally infeasible,
since it would require an infinite amount of filtering and smoothing passes.

Thus, in the setting at hand, $p\left({\mathit{x}}_{n}|{\mathit{y}}_{\mathrm{1}:n}\right)$, the
result of Eq. (16) is given as an infinite mixture of Gaussian mixtures with
weights proportional to *p*(H). Our strategy
of choice to approximately include the systematic uncertainty associated
with **H** is to replace the infinite mixture by a finite version,
i.e., to compute the filtering and smoothing densities for specific
realizations of **H** and combine them into mixtures weighted with the
likelihood of said realization under the density *p*(H) known from prior calibration measurements. In
practice, this is closely related to the selection of sigma points of **H** within
some prior density and using the associated normalized likelihoods as
multiplicative weights with the respective filtering and smoothing mixture
weights to obtain extended mixtures that approximately reflect the
uncertainty associated with **H**, a method inspired by the unscented Kalman filter (Julier et
al., 2000) for approximate filtering in non-linear systems.

## 2.7 Implementation details

Implementation of the presented algorithms was carried out in Python using
the JAX framework (Bradburry et al., 2018), which provides automatic batching
and vectorization, just-in-time compilation, and automatic forward- and
reverse-mode differentiation. This allows the optimization of the
hyperparameters *γ*, *σ*, and **Π** using the
ADAM gradient descent minimization (Kingma and Ba, 2014) routine from JAX
together with the automatically computed batch gradient of the negative
log likelihood, $-{\sum}_{n}\mathrm{ln}\left({\mathit{y}}_{n}|{\mathit{y}}_{\mathrm{1}:n-\mathrm{1}},\mathit{\sigma},\mathit{\gamma},\mathbf{\Pi}\right)$, of the GSF forward pass on the entire
dataset (Algorithm 1 in Appendix A2). The number of components retained in
the arising Gaussian mixtures in GSF and EC is chosen according to available
computational resources. Forward and backward step routines for the linear
dynamical system given in Eq. (2) (cf. Appendix A1) were implemented as
shown in the auxiliary functions for Algorithms 1 and 2 in Appendix A2, respectively. Gaussian mixture reduction as necessary in the GSF forward
and EC backward passes (i.e., the approximation steps in the charts in Figs. 1 and 2) was
implemented according to a greedy algorithm (based on
Kullback–Leibler divergence (Runnalls, 2007). The mean approximation in the
EC backward pass, and implementation of the EC backward and GSF
forward passes, were directly adopted as presented in Barber (2006).
Gaussian mixture weights and model index probabilities are stored and
operated on in ln-space (using the log-sum-exp operation for normalization) for
improved numerical stability. Routines for the computation of the Matrices
**F**, **U**, **M**, **C**, and **B** in Eq. (13) were obtained from symbolic computation using SymPy (Meurer et al., 2017),
and the symbolic Jordan canonical form of **F** and were
hardcoded afterward.

Data for this experiment were generated using an electroplated ^{226}Ra
(104.4±0.4) Bq source (Mertes et al., 2020) mounted on an
electrically cooled high-purity germanium (HPGe) detector placed inside a 20 m^{3} climate chamber. *γ*-ray spectra were recorded
over approximately 85 d at intervals of 10 800 s real time. Within the
time series, regions of missing measurements are present, i.e., *δ*
values varied. At specific times the relative humidity inside the climate
chamber was varied with the intention of inducing changes in the emanation
characteristics of the ^{226}Ra source. Inside the chamber, relative
humidity and temperature were measured in close proximity to the source with
a SHT-35 sensor (Sensirion).

For each spectrum, counts above 200 keV were summed, a lifetime scaled
background count rate (with associated uncertainty that defines
**R**) was subtracted, and the final algorithms described
above (as given in Appendix A2) were applied to the resultant time series of
count values (i.e., 1D measurement series) as the input data
*y*_{1:N}, with five Gaussian components per filter in the GSF forward
and EC backward passes. Results are illustrated for the filtering in Fig. 1a
and for the smoothing in Fig. 1b. Each *r* was chosen to reflect the
real time of each spectrum as provided by the manufacturer's
data-acquisition software (Genie 2000, Mirion Technologies), and each *δ* was computed from
the recorded time stamps of acquisition start points. The dead time of the
system was accounted for by correcting the derived count values using the
dead-time data as provided by the data-acquisition software. The value of
**H** was determined by measurements of a sealed source of similar type
and geometry as presented in Mertes et al. (2020). The uncertainty
associated with the ^{226}Ra activity known from previous measurements
detailed in Mertes et al. (2020) was encoded in the density of the prior
for the state provided to the algorithm. Apart from the ^{226}Ra activity,
a vague Gaussian prior with a diagonal covariance matrix was chosen for
*p*(*x*_{0}). Inherently, the model formulation assumes perfect
equilibrium between the SLP and ^{222}Rn in the source, which is a small
approximation on the timescales at hand. The threshold of 200 keV was
chosen because ^{226}Ra emits *γ* radiation almost entirely below
this energy level, such that the spectrum beyond is almost entirely made up
of events associated with the SLP in the source and the background
radiation. We chose to neglect the information gained from the spectra
regarding the ^{226}Ra activity because it was not found to substantially
improve the prior. The summation of spectra is the most straightforward way
to utilize the information contained within each spectrum while keeping the
dimensionality of ** y** as small as possible. As a result, the prior
density for the

^{226}Ra activity component of the state is retained over the entire dataset, which is why this component of the state is not shown in Fig 3. The last component of the state vectors, $\frac{\mathrm{d}\mathit{\eta}}{\mathrm{d}t}$, is also not shown in Fig. 3 for visual clarity, since it carries no important information and is merely used as a mathematical tool to specify the properties of the stochastic process of

*η*, the main estimation target of this work. The component $\frac{\mathrm{d}\mathit{\eta}}{\mathrm{d}t}$ is therefore not of any practical meaning.

Both the confidence intervals and the median in Fig. 3 were computed from the
marginal cumulative density of the Gaussian mixtures using numerical
root finding. Additionally, the confidence intervals include a systematic,
Gaussian 1 % uncertainty on the specific value of **H** which was
approximately propagated using the derivation in Sect. 2.6 for five distinct
realizations of **H** (${\mathit{\mu}}_{\mathbf{H}},{\mathit{\mu}}_{\mathbf{H}}\pm {\mathit{\sigma}}_{\mathbf{H}},{\mathit{\mu}}_{\mathbf{H}}\pm \mathrm{2}{\mathit{\sigma}}_{\mathbf{H}}$).

In the present work, we have summarized and explained the limitations of
previously available approaches to estimate ^{222}Rn emanation through
measurements of the short-lived progeny (SLP) retained within the source. As
was derived from first principles, those methods to standardize ^{222}Rn
emanation are limited to sources with stable characteristics within the
operational time and, as such, are generally restricted to use in stable
environmental conditions. To alleviate this shortcoming, here we developed
an alternative approach that directly infers the conditional probability
density for the latent ^{222}Rn emanation term from spectral time series
of the SLP that remains within the source.

During the application of the resultant algorithms to real-world data, the deviations of the steady-state approximation of the previous method (red dots in the second row of Fig. 3) from the estimated true values (black lines in the second row of Fig. 3) become apparent, underpinning the theoretical derivations. In turn, this means that a thorough analysis of data obtained in this way is restricted to models that account for the dynamic nature of the system, which has not previously been reported.

The specific structure of the filtering and smoothing results in the second
row of Fig. 3, showing peaked emanation upon increases in humidity, can
easily be explained physically as follows, which justifies the results of
the applied method. Considering the time series of count data of the SLP
within the source (input data; red dots in the first row of Fig. 3), regions
can be seen where changes are occurring much more quickly than would be
possible based on the well-known radioactive decay kinetics. As was
discussed in Sect. 2.2, the time series of counts is theoretically given
by a discretely sampled convolved version of the emanation. Hence, peaked
emanation must be occurring, such that the observed time series is possible
within the theoretically known decay kinetics. Conversely, the drop in
humidity and thus emanation at approximately 70 d does not show this behavior, and
the observed ingrowth of the counts directly follows the decay kinetics.
Apparently, the behavior depends on the direction of change in the
emanation characteristics. This is explained by the fact that upon an
increase of the effective diffusion coefficient, the source retains more
^{222}Rn atoms than the associated equilibrium value, at which point
increased outflow can occur for quick re-equilibration. With the realistic
assumption of zero back diffusion from the volume into the source, for a
change in the opposite direction, the only way to achieve progeny
equilibrium is the typical ingrowth of ^{222}Rn, which is the exact
behavior shown by the deconvolution result but not by the previous method.
Upon fresh preparation of an emanation source, at which point no SLP is
present in the source but emanation is still considered to be happening,
similar count data to the one past 70 d in Fig. 3 may be observed. This
behavior was not previously discussed in Linzmaier and Röttger
(2013), where the apparent initial drop in emanation (as computed by Eq. 4) resulting from the initial ingrowth of residual ^{222}Rn and SLP was
seemingly considered to be its true temporal characteristic, implying that
Eq. (4) may be applied in such a case.

In constant regimes, results obtained from previously reported methods
converge to the values obtained using the deconvolution approach presented
here, as illustrated at approximately days 60 to 70 and past 90 d of the
data shown in Fig. 3. While the method we present might not seem beneficial
in such constant regimes, the recursive Bayesian approach provides a
computationally convenient, mathematically coherent, and flexible way to
refine the uncertainty upon observation of streaming data (e.g., obtained by
continuous operation of spectrometers) also within constant regimes. As
such, here we report for the first time the application of a method whereby
time series data of an emanation source can be used to derive correct (near)
real time values of the emanation, irrespective of the state of the source.
Specifically, the use case of this method and our initial motivation is the
implementation of surveillance systems for emanation sources based on
spectrometric measurements to improve current state-of-the-art realization,
and especially the dissemination of the unit Bq m^{−3} for ^{222}Rn in the low-level activity
concentration regime. With our contribution, and potential extensions
thereof, these types of systems will be enabled in the future. Moreover,
experimental investigations of the emanation behavior in response to
different environmental conditions are drastically improved by our
contribution.

To obtain approximate filtering and smoothing algorithms in the context of radioactivity measurements, we extended the well-known computational methods for inference in linear dynamical systems (i.e., Kalman filter and Rauch–Tung–Striebel smoother) with a computationally convenient approximation for the observed Poisson statistics and the integrating behavior of the measurements in Sect. 2.3 and 2.4. In doing so, we demonstrated that integrating measurements results in a Gaussian process with certain covariance with the latent continuous state which retains the convenient closed form of filtering and smoothing through conjugacy in such linear dynamical models. As was shown, the integrating measurements lead to additional additive uncertainty depending on the variance of the Brownian motion which we consider an intuitive result. These results were used in order to construct the final switching linear dynamical system inference algorithms applied during the experiments.

Within the recorded time series, distinct domains were observed in response
to the way the humidity in the chamber was modified, which lends itself to
the applied switching dynamical system model, differentiating between stable
and non-stable regimes. This approach allows smaller uncertainty to be
achieved and smoother functional realizations of *η* in the somewhat
stable regimes, but at the same time gives reasonably high uncertainty for
the unstable regimes where the deconvolution result relies on only a few
data points. A simpler modeling approach relying only on a single linear
dynamical system, such as a more classical version of the Kalman filter, cannot produce smooth results for the constant regimes while retaining the
ability to react to steep gradients, since both properties are controlled by
the variance of the Brownian motion. While all obvious switching points
(induced by changes in the relative humidity) within the time series were
captured by the algorithm (third row of Fig. 3), the specific autocorrelation
we chose to regularize *η* leads to smearing of the switching point in
the backward (i.e., the smoothing) pass. This is indicated by the fact that
the model proposes an unstable state of the source even for times before the
humidity has undergone the step changes (i.e., in times before a known change
in the source properties has occurred). Note that this is not the case for
the filtering results. This is can be attributed to the applied symmetric
autocorrelation of *η* (Eq. 7) and its independency from the residual
radon activity, and it may be alleviated by asymmetric autocorrelation or
non-linear models but at a substantially higher computational cost. Whether
our approach translates well to time series of different characteristics
(e.g., drift, smooth changes) is as yet unclear and subject to further
studies. Nonetheless, the model parameters provide a way to tune the
algorithm for different scenarios.

For an approach like the present one to be applicable in metrology,
uncertainty estimates closely related to the guide to the expression of uncertainty in measurement (GUM; Joint Committee for
Guides in Metrology, 2008) are needed. At this point, the GUM is restricted
to static measurements and first steps are being taken for an extension to
dynamic scenarios (e.g., in Eichstädt and Elster, 2012; Elster and Link,
2008; Link and Elster, 2009), where a slightly different formulation for
error propagation has been carried out. In the present case, systematic
contributions to the uncertainty are dominated by the uncertainty of the
measurement mapping through matrix **H**. We provided a computationally
simple approach to approximately propagate this uncertainty across the
filtering and smoothing algorithms with arbitrary precision, given that enough
computational resources are available and the detection system can
reasonably be assumed to be stable in time. Dropping this assumption would
require the approximation of the intractable integration in Eq. (15), e.g.,
through Monte Carlo integration, which was found unnecessary and would have
made the algorithm unsuitable for implementation on low-power, portable
devices.

Assuming that the density $p\left({\mathit{x}}_{n-\mathrm{1}}|{\mathit{y}}_{\mathrm{1}:n-\mathrm{1}}\right)$ is given as $\mathcal{N}\left({\mathit{\mu}}_{n-\mathrm{1}}{\mathbf{\Sigma}}_{n-\mathrm{1}}\right)$, we define the state at the
intermediate time point ${t}_{n-\mathrm{1}}+{\mathit{\delta}}_{n}\phantom{\rule{0.125em}{0ex}}$as *x*_{δ}
for which the following statistics are readily available through the
Chapman–Kolmogorov equation (Särkkä and Solin, 2019).

where **F**_{δ}=e^{Kδ} and ${\mathbf{U}}_{\mathit{\delta}}=\phantom{\rule{0.125em}{0ex}}{\int}_{\mathrm{0}}^{\mathit{\delta}}{\mathrm{e}}^{\mathbf{K}\left(\mathit{\delta}-\mathit{\tau}\right)}{\mathbf{LL}}^{\mathbf{T}}{\mathrm{e}}^{{\mathbf{K}}^{\mathbf{T}}\left(\mathit{\delta}-\mathit{\tau}\right)}\mathrm{d}\mathit{\tau}$, as follows from Eqs. (5) and
(11).

By combination of the definition for ** x**(

*t*) in Eq. (10) and the definition of the measurement process for

**in Eq. (12), the measurement at**

*y**t*

_{n}is given by the following integration. While the integral in Eq. (12) is an ordinary integral (e.g., in the Riemann sense), the integral in Eq. (10) is a stochastic integral in the Itō sense; i.e., the following double integral is the ordinary integral over an Itō integral, for which we assume that

**(**

*x**t*) has continuous sample paths and is square integrable.

The triangular domain of the double integral is swapped to find

from which the full density $p\left({\mathit{x}}_{n-\mathrm{1}},{\mathit{x}}_{\mathit{\delta}},{\mathit{x}}_{n},{\mathit{y}}_{n}|{\mathit{y}}_{\mathrm{1}:n-\mathrm{1}}\right)$ can be computed using the expectation operator and the definition of the variance, where 〈⋅〉 denotes the expectation operator. Moreover, by definition, $\langle \int \mathrm{d}\mathit{\beta}\rangle =\mathrm{0}$ and thus

Under the assumption of independence of the Brownian motion and
*x*_{δ}, using Itō isometry and Fubini's theorem,
it follows that

where $\mathbf{M}={\int}_{\mathrm{0}}^{r}{\mathrm{e}}^{\mathbf{K}s}\mathrm{d}s$ and $\mathbf{B}={\int}_{\mathrm{0}}^{r}{\int}_{\mathit{\tau}}^{r}{\int}_{\mathit{\tau}}^{r}{\mathrm{e}}^{\mathbf{K}\left(a-\mathit{\tau}\right)}{\mathbf{LL}}^{\mathbf{T}}{\mathrm{e}}^{{\mathbf{K}}^{\mathbf{T}}\left(b-\mathit{\tau}\right)}\phantom{\rule{0.125em}{0ex}}\mathrm{d}a\phantom{\rule{0.125em}{0ex}}\mathrm{d}b\phantom{\rule{0.125em}{0ex}}\mathrm{d}\mathit{\tau}$.

Analogously, the cross-covariance between *x*_{n} and
*y*_{n} is given as

with $\mathbf{C}={\int}_{\mathrm{0}}^{r}{\int}_{\mathit{\tau}}^{r}{\mathrm{e}}^{\mathbf{K}\left(r-\mathit{\tau}\right)}{\mathbf{LL}}^{\mathbf{T}}{\mathrm{e}}^{{\mathbf{K}}^{\mathbf{T}}\left(a-\mathit{\tau}\right)}\phantom{\rule{0.125em}{0ex}}\mathrm{d}a\phantom{\rule{0.125em}{0ex}}\mathrm{d}\mathit{\tau}$. The result in Eq. (13) is then obtained by marginalizing the full density over
*x*_{n−1} and *x*_{δ}, which for the Gaussian
density means to simply drop the respective columns and rows. Additionally,
the forward filtering step is obtained by conditioning the resultant
$\mathrm{p}\left({\mathit{x}}_{n},{\mathit{y}}_{n}|{\mathit{y}}_{\mathrm{1}:n-\mathrm{1}}\right)$, Eq. (13), onto the observation of
*y*_{n} using the well-known conditioning formula for the Gaussian
distribution. An implementation for such computation is given by the
function *FWD_STEP* in Appendix A2.

Gamma-ray spectra and environmental data obtained for the experimental section as well as the implementation of the presented algorithms and processing software are available at https://doi.org/10.5281/zenodo.7798458 (Mertes, 2023).

AR and SR acquired funding, supervised and conceptualized this work, and reviewed the draft. FM derived the methodology, designed the models, implemented the software, acquired the experimental data, and prepared the original draft.

The contact author has declared that none of the authors has any competing interests.

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

This article is part of the special issue “Sensors and Measurement Science International SMSI 2021”. It is a result of the Sensor and Measurement Science International, 3–6 May 2021.

The authors thank Alan Griffiths (ANSTO) and Scott D. Chambers (ANSTO) for their comments on the paper.

This project has received funding from the EMPIR programme co-financed by
the Participating States and from the European Union's Horizon 2020 research
and innovation programme. 19ENV01 traceRadon denotes the EMPIR project
reference.

This open-access publication was funded by the Physikalisch-Technische Bundesanstalt.

This paper was edited by Alexander Bergmann and reviewed by two anonymous referees.

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- Abstract
- Introduction
- Theory and derivations
- Application to experimental data
- Discussion and conclusion
- Appendix A: Joint density – derivation of discrete forward step for integrating measurement
- Appendix B: Applied algorithms in pseudo-code
- Code and data availability
- Author contributions
- Competing interests
- Disclaimer
- Special issue statement
- Acknowledgements
- Financial support
- Review statement
- References

^{222}Rn from solid sources containing the isotope

^{226}Ra is presented. Therein, supporting radioactivity measurements of the source are used in conjunction with a theoretical description of the dynamics. For radiation protection and environmental research, reliable and comparable

^{222}Rn measurements, and therefore reference atmospheres of

^{222}Rn, are needed. This work improves their realization.

^{222}Rn...

- Abstract
- Introduction
- Theory and derivations
- Application to experimental data
- Discussion and conclusion
- Appendix A: Joint density – derivation of discrete forward step for integrating measurement
- Appendix B: Applied algorithms in pseudo-code
- Code and data availability
- Author contributions
- Competing interests
- Disclaimer
- Special issue statement
- Acknowledgements
- Financial support
- Review statement
- References