# Many wrong models approach to localize an odor source in turbulence:
introducing the weighted Bayesian update

Lorenzo Piro<EMAIL_ADDRESS>Department of Physics & INFN,
University of Rome “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Rome,
Italy  Robin A. Heinonen  Department of Physics & INFN, University of Rome
“Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Rome, Italy  Massimo
Cencini  Istituto dei Sistemi Complessi, CNR, Via dei Taurini 19, 00185 Rome,
Italy  INFN “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Rome, Italy
Luca Biferale  Department of Physics & INFN, University of Rome “Tor Vergata”,
Via della Ricerca Scientifica 1, 00133 Rome, Italy

###### Abstract

The problem of locating an odor source in turbulent environments is central to
key applications such as environmental monitoring and disaster response. We
address this challenge by designing an algorithm based on Bayesian inference,
which uses odor measurements from an ensemble of static sensors to estimate
the source position through a stochastic model of the environment. Given the
practical impossibility of achieving a fully consistent turbulent model and
guaranteeing convergence to the correct solution, we propose a method to rank
“many wrong models” and to blend their predictions. We evaluate our _weighted
Bayesian update_ algorithm by its ability to estimate the source location with
predefined accuracy and/or within a specified time frame, and compare it to
standard Monte Carlo sampling methods. To demonstrate the robustness and
potential applications of both approaches under realistic environmental
conditions, we use high-quality direct numerical simulations of the Navier-
Stokes equations to mimic the transport of odors in the atmospheric boundary
layer. Despite minimal prior information about the source and environmental
conditions, our proposed approach consistently proves to be more accurate,
reliable, and robust than Monte Carlo methods, thus showing promise as a new
tool for addressing the odor source localization problem in real-world
scenarios.

##  I Introduction

Identifying the origin of noxious odors such as gas leaks and chemical or
radioactive emissions is critical for averting potential environmental
disasters [1, 2, 3, 4]. Moreover, tracing the location of a source from the
odor signal detected in the atmosphere is a vital task for animals looking for
food or a mate [5, 6].

Such source localization problems are already difficult in laminar flows,
where the structure of the odor plume can be highly complex and sensitive to
the source location due to chaotic mixing [7, 8]. In fully turbulent three-
dimensional (3-D) environments, the difficulty becomes severe, owing to
numerical, experimental, and phenomenological complexities and challenges
associated with the multi-scale nature of 3-D turbulence, which is
characterized by a broad range of rough velocity fluctuations as well as
highly intermittent and non-Gaussian properties for both the advecting flow
and the odor transport [9, 10, 11, 12].

This makes the design of efficient and reliable algorithms for source
localization a challenging theoretical task situated at the intersection of
fluid dynamics, optimal navigation of mobile agents, and information theory
[12, 13, 14]. Over the last decades, a growing community of scientists has
developed multiple methodologies to address this problem, ranging from bio-
inspired [15, 16, 17] to heuristic search algorithms based on information
theory [18, 19, 20] and Bayesian inference [21, 22].

Although most studies have focused on the performance optimization of mobile
agents, the use of static sensors has become increasingly popular [23, 24, 25]
thanks to the simplicity of their implementation, also resulting in reduced
setup and maintenance costs. Moreover, being well-suited for continuous and
large-scale environmental monitoring, they are ideal for use in early warning
systems [26, 27, 28, 29], which may enable targeted and effective response
strategies, ranging from containment and mitigation to rescue and prevention.
On a more theoretical level, the employment of static sensors reduces the
space of possible choices for the search algorithm [30], thereby facilitating
the study of more fundamental questions. Indeed, despite the recent progress,
the problem of optimally locating a source within a given time with a
prescribed accuracy in a turbulent flow is far from solved [25].

Figure 1: Left: a snapshot of DNS showing relative positions of sources
(squares) and particles color-coded according to the source from which each
was emitted. Here, we show particles in a thin slab parallel to the wind and
containing the sources, and we have zoomed in close to the sources. We have
also inserted in the same plane a network of static sensors (yellow circles)
whose measurements will be used in the following to infer the source location.
Right: probability map of making a detection. Hereafter referred to as
_empirical likelihood_ , such distribution has been obtained from coarse-
graining the DNS data on a square lattice, setting a threshold $n_{\rm thr}=2$
on the number of detectable particles, and then averaging over time the signal
of all the five sources (properly placed, by means of a suitable shift, in the
same position as indicated by the grey square).

A key challenge is the practical impossibility of specifying a detailed model
for the statistics governing odor transport. This is a crucial point when
using Bayesian approaches that heavily depend on a model of the environment to
integrate sensor observations into a probability distribution describing our
information about the source. However, in all real-world applications, we have
only partial knowledge of the statistical properties of the environment, which
may be arbitrarily complex. Moreover, the inherent complexity of turbulent
transport precludes a complete analytical description even in an idealized
physical scenario. Therefore, we are invariably compelled to work with an
_wrong_ model of the environment. This fundamentally limits the utility of
Bayesian approaches, which are only guaranteed to converge to the ground truth
when the model is correctly specified [31, 32]. To partially overcome these
intrinsic limitations, we explore a new Bayesian approach designed to mitigate
the effects of errors in the environmental model. Building upon the _many
wrongs principle_ [33, 34], we introduce a quantity that allows to rank the
models of the environment in the Bayesian framework, providing a principled
way to blend information coming from several (inevitably wrong) models. We
show that merging the information gathered from a number of different models –
a procedure that we dubbed Weighted Bayesian Update (WBU) – helps in reliably
inferring the source location.

In order to test these ideas and the proposed methodology in realistic
conditions, we devise a set of direct numerical simulations that mimic the
emission of a localized odor source in a three-dimensional turbulent
environment. Taking advantage of the Galilean invariance of the Navier-Stokes
equation, we model the odor transport by Lagrangian tracers advected by the
flow with a constant mean wind, obtaining odor plumes that well approximate
those one can observe in the atmospheric boundary layer [11, 12]. Here, for
the sake of a first benchmark of the localization algorithms, we shall assume
the source lies on the same plane as the sensors, which can thus detect odor
particles contained in a thin slab parallel to the wind [see Fig.1(a)]. Then,
we use a simplified (effective) model for the environment that depends on a
few parameters only and neglects temporal and spatial correlations of the odor
signal. In spite of the obvious shortcomings of this model, we show how, by
suitably weighting different models (i.e. the same one with different
parameters), WBU outperforms widely-used Monte Carlo methods [21, 35] when
locating the odor source with a specified level of accuracy. By performing
several tests with synthetic data and empirically based (_a posteriori_)
models, we trace the origin of the problems for Monte Carlo methods to the
effects of correlations in the realistic odor signal, to which the WBU shows
superior resilience.

The paper is organized as follows. In Sec. II, we illustrate the setup where
the numerical simulations have been carried out and describe the simplified
model of the environment as well as the algorithms employed, namely the
weighted Bayesian update and another approach based on Monte Carlo techniques.
We then present the results in Sec. III, which is divided into two parts. In
the first one, Sec. III.1, we discuss and compare the performance of the
algorithms in the setting closest to reality. In Sec. III.2, we then
systematically study how model inaccuracy affects the quality of the source
localization. Finally, we summarize our findings and provide an outlook for
future studies in Sec. IV.

##  II Methods

###  II.1 Setup of the numerical simulations

Let us assume we aim to locate the position $\bm{r}_{\rm s}$ of an odor source
emitting at rate $Q$ within a two-dimensional square grid $\Omega$ of size
$L\times L$ and lattice spacing $\Delta x$. The odor particles are swept by
the underlying turbulent flow characterized by a mean wind in a given
direction $\hat{\bm{U}}$. We then place a number $N_{\rm s}$ of static sensors
in this environment. We arrange the sensors in a lattice that, assuming no
prior knowledge of the wind direction, will be typically tilted by an angle
$\theta$ with respect to the latter, as depicted in Fig. 1(a).

Instead of using an odor concentration field, we model it in terms of
particles advected by a turbulent flow. Each particle can be thought of as a
patch of odor, or one can consider the number of particles in a given small
region as an estimate of the odor concentration. We produced realistic
trajectories of odor particles using state-of-the-art direct numerical
simulation (DNS) of the incompressible, 3–D Navier-Stokes equations

$\displaystyle\partial_{t}\bm{u}+(\bm{u}\cdot\nabla)\bm{u}$
$\displaystyle=-\nabla p+\nu\nabla^{2}\bm{u}+f,$ (1)
$\displaystyle\nabla\cdot\bm{u}$ $\displaystyle=0,$ (2)

under turbulent conditions with $\mathrm{Re}_{\lambda}\simeq 150.$ Here, $f$
is a random, Sawford-type [36] isotropic forcing at the smallest nonzero
wavenumbers of the system, with a correlation time of 160 simulation
timesteps. Using a pseudospectral code dealiased according to the two-thirds
rule, the system was solved on a $1024\times 512\times 512$ grid, with a
uniform spacing $\delta x=\delta y=\delta z\simeq\eta$ (with $\eta$ the
Kolmogorov scale), and periodic boundary conditions in all three directions.
The timestepping was performed using the second-order explicit Adams-Bashforth
method. The system was advected by a uniform mean wind
$\bm{U}\approx-2.5u_{\rm rms}\hat{x},$ where $u_{\rm rms}$ is the rms speed of
the flow in the comoving frame of the wind, and $\hat{x}$ is the elongated
axis of the grid. We produced the mean wind by means of a Galilean
transformation.

The odor particles were modeled as Lagrangian tracer particles, which were
emitted by five stationary point sources [Fig. 1(a)]. Each source emitted 1000
particles every 10 simulation timesteps, which corresponds to every $\approx
1/15$ Kolmogorov times $\tau_{\eta}$. The fluid velocities $\bm{u}$ at the
particle positions were obtained using a sixth-order B-spline interpolation
scheme and then used to evolve the particle positions $\bm{X}$ in time
according to $\dot{\bm{X}}=\bm{u}(\bm{X},t)$ over an infinite lattice of
copies of the periodic flow. Their positions, velocities, and accelerations
were tracked and dumped every $\tau_{\eta},$ for a total of 3015 timesteps.
Each source of particles was treated as independent, and we averaged our
results over them for the purpose of achieving better statistics.

To simulate realistic environmental conditions and emulate turbulent
dispersion in the atmospheric boundary layer, we then coarse-grain the
particles’ concentration inside a thin layer containing the source and set a
threshold $n_{\rm thr}=2$ on the number of particles above which sensors make
a detection. A few snapshots of the resulting odor dispersion in the
aforementioned arena $\Omega$ are shown in Fig. 2, where the pink patches
indicate the odor plume.

Figure 2: Three time snapshots showing a qualitative comparison between the
performance of the two algorithms discussed in this work. Both approaches use
the measurements made by an array of static sensors (circles) looking for a
source of odors (red square) advected by a turbulent flow featuring a
horizontal wind blowing from right to left. Pink patches indicate the odor
plume; greyscale codes the probability of the odor source location obtained
from an algorithm based on Bayesian inference; green open triangles depict the
candidate source positions yielded by a Sequential Monte Carlo sampling
method. Time here is in units of observations made by each sensor, which is
hereafter assumed to be equal to the Kolmogorov timescale $\tau_{\eta}$ of the
flow.

###  II.2 Model of the environment

To localize the source within this setting given the sensors’ measurements, we
shall now introduce a model that captures the mean-flow and mean-diffusion
properties of the environment. If, on the one hand, we assume to have a
statistical knowledge of the environment through, for example, the history of
prior measurements in the field, we can compute empirically what is the
probability of detecting an odor particle from the time average of the DNS
data [Fig. 1(b)]. We will hereafter refer to this distribution as the
_empirical likelihood_ , in accordance with Bayesian nomenclature. The use of
such likelihood has the advantage of avoiding the complication of fitting some
environmental parameters while looking for the source. However, it relies on a
far more detailed prior knowledge of the environment, which is not generally
available. On the other hand, we can still exploit it to understand the
strengths and weaknesses of the localization algorithms, as discussed in Sec.
III.2.

In a realistic setup, we are therefore compelled to use a statistical
description of odor encounters in a turbulent flow to infer the source
location. Out of a wide range of possibilities [37], we model the turbulent
transport of odor particles emitted at rate $Q$ by a point source as an
effective advection-diffusion process [38]

$\partial_{t}c+\bm{U}\cdot\bm{\nabla}c=D\nabla^{2}c+Q\delta(\bm{r}-\bm{r}_{\rm
s})-c/\tau\,,$ (3)

where $c$ is the odor concentration field, $\bm{U}$ the mean wind featured by
the turbulent flow, $\tau$ the lifetime of the odors, and $\bm{r}_{\rm s}$ the
source position. Note that the combination of molecular and turbulent
diffusivity (due to the flow velocity fluctuations) is here described by a
single effective (eddy) diffusion coefficient $D$ [39]. Despite being a strong
oversimplification that ignores important multiscale, non-Gaussian properties
of the underlying turbulence, as well as spatiotemporal fluctuations in the
scalar advection, this model reasonably captures the mean field properties of
the odor concentration [12]. In the stationary regime, Eq. (3) has an
analytical solution, which in three dimensions reads (see, e.g., [18])

$c(\bm{r}-\bm{r}_{\rm s})=\frac{Q}{4\pi D\lVert\bm{r}-\bm{r}_{\rm
s}\rVert}\exp\left[\frac{(\bm{r}-\bm{r}_{\rm
s})\cdot\hat{\bm{U}}-\lVert\bm{r}-\bm{r}_{\rm s}\rVert}{\lambda}\right]\,,$
(4)

where we have assumed $\tau\gg 1$. A key advantage of the adopted model is
that it essentially depends only on three environmental parameters, i.e., the
source emission rate $Q$, the mean wind direction $\hat{\bm{U}}$, and the
characteristic length scale of the flow $\lambda\equiv 2D/U$. To mimic the
sparseness and intermittency of the odor signals observed from the DNS, and
typical of turbulent environments, we assume that the detection is a random
process modeled in terms of a Poissonian process with mean $\mu$ [18, 40].
Therefore, the probability that a sensor makes a detection is given by

$p(h_{i}|\bm{r}_{i}-\bm{r}_{\rm s})=\frac{[\mu(\bm{r}_{i}-\bm{r}_{\rm
s})]^{h_{i}}\exp[-\mu(\bm{r}_{i}-\bm{r}_{\rm s})]}{h_{i}!}\,,$ (5)

where $h_{i}$ is the number of odor particles detected by the $i-$th sensor.
The mean number $\mu$ of particles hitting, within a time interval $\Delta t$,
the $i$-th sensor is related to the mean concentration (4) via the classical
Smoluchowski formula [41]

$\mu(\bm{r}_{i}-\bm{r}_{\rm s})=4\pi aD\Delta t\,c(\bm{r}_{i}-\bm{r}_{\rm
s})=\frac{Q}{d_{i}}a\Delta t\exp\left[\frac{(\bm{r}_{i}-\bm{r}_{\rm
s})\cdot\hat{\bm{U}}-d_{i}}{\lambda}\right]\,,$ (6)

where $\bm{r}_{i}$ stands for the $i-$th sensor position, and
$d_{i}\equiv\lVert\bm{r}_{i}-\bm{r}_{\rm s}\rVert$, $a$ is the sensors’
radius. Hereafter, we will actually assume that each sensor can detect the
presence of odors only if the number of particles within its radius $a=\Delta
x/2$ exceeds a particular threshold. In other words, the sensors can only
perform binary measurements (i.e., $h_{i}=\\{0,1\\}$), such that the
probability of detection (5) simplifies into

$\begin{cases}p(0|\bm{r}_{i}-\bm{r}_{\rm s})=\exp[-\mu(\bm{r}_{i}-\bm{r}_{\rm
s})]\\\ p(1|\bm{r}_{i}-\bm{r}_{\rm s})=1-\exp[-\mu(\bm{r}_{i}-\bm{r}_{\rm
s})]\,.\end{cases}$ (7)

###  II.3 Weighted Bayesian Update: a new way to rank and exploit _many wrong_
models

Each measurement made by each sensor thus provides information about the
position of the source $\bm{r}_{\rm s}$, which can be processed employing
Bayesian inference. The whole set of measurements from all sensors can be used
to update a probability map – the “posterior” or “belief” in Bayesian jargon –
of the source’s location defined over the whole arena $\Omega$, i.e.
$B(\bm{r})\equiv\mathrm{Prob}(\bm{r}_{\rm s}=\bm{r})$, which we shall dub as
_common belief_ to emphasize that it exploits all sensors’ detections. Since
we assume no prior knowledge and that the source cannot be at the same
location as one of the sensors, the belief is always initialized to a uniform
distribution and set to zero only in the sensors’ positions. Assuming the
simultaneous measurements made by all the $N_{\rm s}$ sensors at time $t$ are
independent, the overall conditional probability of a set of observations
$\bm{h}^{(t)}$ for a possible given source position $\bm{r}$ is simply a
product:

$\mathcal{L}(\bm{h}^{(t)}|\bm{r})=\prod\limits_{i=1}^{N_{\rm
s}}p(h_{i}^{(t)}|\bm{r}_{i}-\bm{r})\,,$ (8)

which depends on the model of the environment as specified in Eq. (7), and is
also known as the _likelihood_ function in Bayesian terminology. Given a
sequential process like the one at hand, at every time step (i.e., once all
sensors have performed a measurement), the belief is updated following Bayes’
rule [42]

$B^{(t)}(\bm{r})=\frac{\mathcal{L}(\bm{h}^{(t)}|\bm{r})B^{(t-1)}(\bm{r})}{\int_{\Omega}{\rm
d}\bm{r}\,\mathcal{L}(\bm{h}^{(t)}|\bm{r})B^{(t-1)}(\bm{r})}\,,$ (9)

where time $t$ is hereafter measured in units of observations made by each
sensor.

Under this update rule, the belief is guaranteed to converge to the correct
solution as long as the correct model of the environment is deployed [43].
This is, however, a rather uncommon case in any realistic scenario, and the
model used to update the common belief will always be wrong. Furthermore,
wrong models are indistinguishable from one another during the search since
they always make the belief converge to a source position, regardless of
whether that is the right one or not. Therefore, finding a way to rank the
models and assess their reliability is of great relevance for any
applications.

In order to introduce the quantity we used for ranking the models, it is
useful to rewrite Bayes’ update rule (9) in a different way:

$B^{(t)}(\bm{r})=\frac{\prod\limits_{i=1}^{N_{\rm
s}}b_{i}^{(t)}(\bm{r})}{\mathcal{Z}}\,,\;\;\;\;\;\;b_{i}^{(t)}(\bm{r})=\frac{p(h_{i}^{(t)}|\bm{r}_{i}-\bm{r})b_{i}^{(t-1)}(\bm{r})}{\int_{\Omega}{\rm
d}\bm{r}\,p(h_{i}^{(t)}|\bm{r}_{i}-\bm{r})b_{i}^{(t-1)}(\bm{r})}\,,$ (10)

where $\mathcal{Z}\equiv\int_{\Omega}{\rm d}\bm{r}\,\prod\limits_{i=1}^{N_{\rm
s}}b_{i}^{(t)}(\bm{r})$. That is to say, owing to the assumption of
independence between measurements, the common belief can be built from the
superposition of the _private_ beliefs (i.e. the belief that can be
constructed for each sensor only on the basis of its own measurements history)
$b_{i}$ of the $i=1,...,N_{\rm s}$ sensors, each of which is updated
independently at every time step via Bayes’ rule. The key advantage of this
change of perspective lies in the interpretation of the update formula [Eqs.
(9-10)]. Indeed, the hitherto overlooked normalization constant $\mathcal{Z}$
essentially quantifies how much the sensors’ _private_ beliefs overlap or, in
other words, how much they agree on a source position given their measurements
and the model at hand. In fact, in Appendix A, we show analytically that this
quantity achieves an asymptotic global maximum when the model is exact (even
in the presence of correlations between observations). We also show
numerically that this quantity is typically smaller the farther the model is
from the ground truth.

This makes $\mathcal{Z}$, hereafter referred to as the _overlap integral_ , an
ideal candidate to achieve our task as we could use it to weigh different
models of the environment. To this end, let us scan the parameters space of
the stochastic model introduced above [see Eqs. (6)-(7)] assuming to know only
on the mean wind direction $\hat{\bm{U}}$, which can be always measured in
practice using an anemometer, and run the Bayes update (9) independently for
each set of parameters $\bm{P}\equiv\\{Q,\lambda\\}$. Then, we shall blend the
information collected into a master belief as

$\mathcal{B}^{(t)}_{\rm
M}(\bm{r})=\frac{\prod\limits_{j=1}^{M}\left[B_{j}^{(t)}(\bm{r})\right]^{\beta_{j}^{(t)}}}{\int_{\Omega}\prod\limits_{j=1}^{M}\left[B_{j}^{(t)}(\bm{r})\right]^{\beta_{j}^{(t)}}}\,,\;\;\;\beta_{j}^{(t)}\equiv\frac{\mathcal{Z}_{j}^{(t)}}{\sum\limits_{j=1}^{M}\mathcal{Z}_{j}^{(t)}}\,,$
(11)

where $M$ is the total number of distinct sets
$\bm{P}_{j}=\\{Q_{j},\lambda_{j}\\}$ of model parameters considered, and the
index $j$ refers to which of these sets was used to obtain the common belief
$B_{j}^{(t)}$ at time $t$ and the corresponding overlap integral
$\mathcal{Z}_{j}^{(t)}$.

The algorithmic procedure described in Eqs. (9)$-$(11) thus provides a
principled way to define a single master belief $\mathcal{B}_{\rm M}$, i.e.,
the probability distribution about the source location, which is obtained from
a weighted average over the results yielded by different models. We will
hereafter refer to this approach as the weighted Bayesian update (WBU), whose
numerical implementation is detailed in Appendix B.

###  II.4 Sequential Monte Carlo with Importance Sampling

The above-described WBU method offers a new perspective on the implementation
of algorithms for odor source localization in turbulent flows. There exists,
however, already a vast literature on the possible methods to address the same
kind of problem [25]. One of the most commonly used approaches is the one
based on the use of Monte Carlo sampling methods to estimate the belief [21,
40, 35]. It is, therefore, natural to ask how the WBU’s performance compares
with such more conventional approaches.

Due to the wide range of potential applications [44], much research has been
conducted over the last decades to make these algorithms more and more
computationally efficient and accurate, and, to date, there exist many
different variants depending on the specific setup at hand. We refer the
reader to Ref. [25] for a comprehensive review of the topic.

In the following, we shall use a state-of-the-art version of the so-called
_Sequential Monte Carlo_ (SMC) algorithm that also involves a Markov chain
Monte Carlo (MCMC) perturbation step [45, 46]. Using such an approach, we can
simultaneously infer both the source position $\bm{r}_{\rm s}$ and the
(unknown) parameters $\bm{P}$ of the stochastic model of the environment. Let
us therefore first define a _sample_ $\bm{\theta}_{i}$ as a possible
combination of such source parameters, i.e.
$\bm{\theta}_{i}\equiv\\{\bm{r}_{{\rm s},i},\bm{P}_{i}\\}$. At every time
step, after all sensors have measured, a collection of $N$ of such samples is
drawn from the current belief defined in the $\bm{\theta}$ space, and we
assign a weight $w$ to each of them equal to the likelihood of the latest
measurement as defined in Eq. (8). Upon normalization of the samples’ weights,
we shall then compute the effective sample size $N_{\rm eff}$ to avoid the so-
called _degeneracy problem_ [46]. Indeed, if $N_{\rm eff}$ goes below a given
threshold $N_{\rm thr}$ (typically set to $N/2$), then it is necessary to
generate a new set of $N$ samples. This is known as the _resampling_ step.
There, for each sample $\tilde{\bm{\theta}}_{i}$, a new one is selected from
the same pool and with a probability equal to its weight that replaces the
former. After resampling, all weights are then set equal to $1/N$.

Next is the so-called Metropolis-Hastings MCMC perturbation step, which
consists of moving each sample in its neighborhood and deciding whether to
accept or reject the new proposal based on some acceptance criteria [47]. This
is essentially done to diversify the $N$ samples and, therefore, improve the
sampling efficiency of the Monte Carlo algorithm [46]. More specifically,
starting from one of the samples at time $t$, say
$\tilde{\bm{\theta}}_{i}^{(t)}$, a Markov chain of length $K$ is generated
where new inferences $\hat{\bm{\theta}}^{(t)}$ are drawn from the previous
link in the chain, $\tilde{\bm{\theta}}_{i,j-1}^{(t)}$, using a proposal
distribution $q(\hat{\bm{\theta}}^{(t)}|\tilde{\bm{\theta}}_{i,j-1}^{(t)})$.
Although there exist several valid choices for this distribution [48], here we
shall use a Gaussian with a mean $\mu=\tilde{\bm{\theta}}_{i,j-1}^{(t)}$ and
variance $\sigma^{2}$ as a free hyperparameter, i.e.
$q(\hat{\bm{\theta}}^{(t)}|\bm{\theta}^{(t)}_{i,j-1})=\mathcal{N}(\bm{\theta}^{(t)}_{i,j-1},\sigma^{2})$.
Once the new sample is generated, we shall compute the so-called _acceptance
ratio_ [45]

$\alpha=\prod\limits_{k=1}^{t}\left[\frac{\mathcal{L}(\bm{h}^{(k)}|\hat{\bm{\theta}}^{(k)})}{\mathcal{L}(\bm{h}^{(k)}|\bm{\theta}^{(k)}_{i,j-1})}\right]\frac{q(\bm{\theta}^{(t)}_{i,j-1}|\hat{\bm{\theta}}^{(t)})}{q(\hat{\bm{\theta}}^{(t)}|\bm{\theta}^{(t)}_{i,j-1})}\,,$
(12)

which basically amounts to the likelihood’s history ratio (also known as the
_posterior ratio_) divided by the proposal ratio. Including the latter is
necessary to correct a bias in the proposal distribution if it is asymmetric
[45]. The proposed sample $\hat{\bm{\theta}}^{(t)}$ will then be accepted as a
new link in the chain as long as $\alpha>1$, and with probability $\alpha$
otherwise.

Finally, once all samples have been perturbed, we are left with a new
approximation of the belief, which reads:
$\tilde{B}^{(t)}(\bm{\theta})=1/N\sum_{i=1}^{N}\delta(\bm{\theta}-\bm{\theta}^{(t)}_{i})$.
A detailed step-by-step description of the procedure outlined above is given
in Appendix B.

Analogous to what we observed in the Bayesian update, once the SMC algorithm’s
hyperparameters are properly adjusted, it will systematically converge to the
correct source location as long as the model of the environment is
functionally exact. However, it is not clear how this approach would compare
with the WBU one in a realistic scenario where it uses an inevitably
misspecified model of the environment.

##  III Results

###  III.1 Stop criterion for source localization

The performance of localization algorithms is typically judged based on their
ability to estimate the odor source position with a specified accuracy or
within a given time limit [25, 49, 50]. However, more generally, we shall test
the reliability of such algorithms by envisioning their application in a real-
world scenario, where we do not know _a priori_ where the source is and have
to decide when to stop looking for it. Therefore, for an algorithm to be
effective in practice, it must show a correlation between an observable
quantity and the quality of its estimate of the source location.

Figure 3: Four panels showing the probability that the distance $d$ between
the estimated source position $\bm{\bar{r}}$ and the ground truth $\bm{r}_{\rm
s}$ is below a given threshold $\delta$ (indicated in the legend) as a
function of the belief/sample standard deviation $\sigma_{\rm loc}$. Lengths
are in units of the distance between sensors. The columns differ in the number
of sensors, while the two rows correspond to the algorithm used (top: WBU,
bottom: SMC).

To this end, a good candidate quantity is the current uncertainty one has
about the source location. This can be formally defined in the WBU approach as
the variance of the master belief (11), which reads

$\sigma^{2}_{\rm loc}\equiv\int_{\Omega}\,{\rm d}\bm{r}\,\mathcal{B}_{\rm
M}(\bm{r})(\bm{r}-\bm{\bar{r}})^{2}\,,$ (13)

where $\bm{\bar{r}}\equiv\int_{\Omega}\,{\rm d}\bm{r}\,\mathcal{B}_{\rm
M}(\bm{r})\bm{r}$ is the estimated source location 111Clearly, the
$\sigma^{2}_{\rm loc}$ and $\bm{\bar{r}}$ counterparts in the SMC algorithm
are respectively defined as the variance and mean computed over the Monte
Carlo samples..

In the ideal case where the model of the environment is exact,
$\sigma^{2}_{\rm loc}$ would be inversely correlated with how close the
estimated position is to the ground truth. In other words, the smaller the
variance of the belief, the more accurate the source location estimate would
be. However, the exact model of the environment is inaccessible to us in any
practical scenario. Therefore, such correlation is not guaranteed to hold in
general and may depend on the algorithmic procedure employed.

We shall thus quantify the robustness of a given localization algorithm by
looking at the probability $P(d<\delta|\sigma_{\rm loc})$ that the distance
$d$ between $\bm{\bar{r}}$ and the actual source position $\bm{r}_{\rm s}$ is
below a given threshold $\delta$, conditioned on the uncertainty on the source
estimate as defined in (13). In particular, the reliability of a localization
algorithm can then be measured by checking whether such probability
$P(d<\delta|\sigma_{\rm loc})$ correlates with the standard deviation
$\sigma_{\rm loc}$.

We find that this is indeed the case for the WBU approach when implemented on
the most realistic scenario introduced in Sec. II.1. As shown in Fig. 3, the
probability of locating the source within a distance smaller than the one
between sensors is well-correlated with the corresponding uncertainty when
using WBU (top row). However, this does not hold for the more standard SMC
algorithm, which hardly shows any correlation (bottom row). Moreover, this
turns out to be independent of the number of sensors deployed ($N_{\rm
s}=\\{16,64\\}$ are shown in the same figure) and of the threshold $\delta$ on
the accuracy (different symbols in each plot). We refer the reader to Appendix
B for further details on the numerical simulations.

Hence, our results suggest that the WBU is a more reliable method than the
standard SMC in the sense that, in WBU, the uncertainty about the source
location $\sigma_{\rm loc}$ is a more reliable indicator of the proximity of
its estimate to the ground truth than it is in SMC.

Furthermore, this has the additional benefit that it allows us to define a
robust stop criterion. Indeed, when employing WBU, it will be sufficient to
look at the time evolution of $\sigma_{\rm loc}$ to know when one has a good
chance of finding the source at a sufficiently small distance from the current
estimate and stop the search accordingly.

Figure 4: Same as Fig. 3 but computed using the models which ranked first with
the overlap criterion discussed in Sec. II.3.

Remarkably, this would no longer hold if we used only the model with the
largest overlap integral $\mathcal{Z}$ to infer the source location. Indeed,
in such a case, the correlation between the distance from the true source
location and its uncertainty almost disappears, with the outcome comparable to
the one of the SMC algorithm, compare Fig. 4 with the top panels of Fig. 3.
This emphasizes the importance of blending information from many wrong models
to estimate the source location, as done in the WBU approach.

###  III.2 Model misspecification: the effect of correlations

Although WBU and SMC use both the same (wrong) model of the environment to
infer the source location, they have different outcomes, with the former
proving to be less sensitive to the model misspecification than the latter. It
is, therefore, worth investigating more systematically how model errors affect
the performance of such source localization algorithms. As shown in the table
in Fig. 5(a), we can consider four possible ways to infer the source location
with a given algorithm when dealing with realistic data of turbulent odor
dispersion.

The first possibility is to use the Lagrangian time series obtained from the
DNS to determine whether a sensor detects an odor or not and then deploy a
(inevitably misspecified) model of the environment —like the one derived from
Eq. (3)— to interpret such detections and compute the likelihood (_Model
likelihood-Correlated signal_ —MlCs— top left cell in the table). This is
precisely the scenario we have analyzed so far, which is the closest one to
reality as it relies on minimal prior knowledge of the environment and
directly uses the Lagrangian data.

Alternatively, as a model of the environment, we can use the empirical
likelihood [Fig. 1(b)] computed from the time history of all the odor
trajectories obtained from the DNS. On the one hand, this approach [_Empirical
likelihood-Correlated signal_ —ElCs— bottom left cell in the table in Fig.
5(a)] greatly simplifies the search since there are no environmental
parameters to fit, and it also represents the best existing (empirical) model
one can aim for in practice to infer the source location. On the other hand,
however, it is still an imperfect model of the environment as it does not
capture the time and spatial correlations featured by the odor plume [12].

To study how such correlations affect the source position estimation, we
should then consider the case where the detections are not directly taken from
the DNS time series but instead randomly drawn from the empirical likelihood.
Indeed, the sensors’ measurements are in this way uncorrelated while still
featuring the same detection statistics as the original signal. At this point,
we can decide whether to deploy the empirical likelihood itself to infer the
source location, in which case we would be using the exact model of the
environment (_Empirical likelihood-Uncorrelated signal_ —ElUs— bottom right
cell in the table) or the usual probabilistic model of detections (_Model
likelihood-Uncorrelated signal_ —MlUs— top right cell in the table). The
former can be helpful as a benchmark since, provided the number of
measurements is large enough, it should show convergence to the correct
solution for any properly implemented localization algorithm based on Bayesian
inference. The latter is instead useful compared to the first two cases
illustrated above since, in this case, sensors’ observations are uncorrelated
and, as a result, it isolates the effect of dealing with a functionally wrong
model of the environment.

Figure 5: (a) Table summarizing all the possible ways to infer the source
location when dealing with realistic data of turbulent odor dispersion. They
may differ for the model used to compute the likelihood (rows) or for the
signal received by the sensors (columns). (b) Mean distance $d$ from the
source at the end of each episode in units of the distance $d_{\rm{s}}$
between sensors as a function of the number of sensors $N_{\rm s}$. Different
symbols correspond to the algorithm employed (SMC: inverted purple triangles;
WBU: yellow diamonds). Error bars are the first and third quartiles of the
distribution. Each plot refers to one of the four cases illustrated in table
(a), as indicated by the box color in the upper left corner.

This completes the picture in the table in Fig. 5(a). We are now ready to
compare the performance of WBU and SMC in each of the four scenarios just
described. To this end, we shall look at the results reported in Fig. 5(b).
There, we show the average distance $d$ between the actual source location and
its estimate given by either algorithm at the end of distinct episodes of set
time length as a function of the number of sensors $N_{\rm s}$ deployed in the
search (details in Appendix B). Some observations are in order. First of all,
both WBU (yellow diamonds) and SMC (purple inverted triangles) show comparable
performance as long as they use the empirical likelihood to model the
detection statistics (plots in the bottom row). More specifically, when using
the exact model as in the ElUs scenario, both approaches tend to converge to
the correct source location as expected (bottom right plot). At the same time,
by looking at the ElCs case (bottom left plot), we may observe how much the
sole presence of time and spatial correlations in the signal affects the
quality of the source estimate. Remarkably, even though the empirical model
does not account for such correlations, both algorithms manage to locate the
source within a distance $d$ smaller than half of the sensor separation.

Now, let us see what changes when using the probabilistic model introduced in
Sec. II.2 to infer the source location (MlCs and MlUs scenarios —plots in the
top row). While using the wrong model has obvious shortcomings and further
degrades the performance of both approaches, overall, they prove to be quite
robust and still manage, on average, to locate the source within a distance
smaller than the one between sensors. Furthermore, comparing the results
obtained with uncorrelated signals (MlUs, top right plot) with the most
realistic scenario featuring the correlated signal (MlCs, top left plot), we
may notice that the SMC’s performance gets substantially worse in the latter
case, while WBU is basically unaffected.

The higher sensitivity of SMC to correlations can be rationalized by looking
at the time evolution of the model parameters, namely the emission rate $Q$
and the characteristic length scale of turbulent odor dispersion $\lambda$,
inferred by such an algorithm. As shown in Fig. 6, when the signal received by
the sensors is correlated in space and time (blue curves), SMC has the
tendency to overestimate both $\lambda$ and $Q$, even saturating to the
maximum allowed value. The picture dramatically changes when the sensors’
observations are instead uncorrelated (orange curves). There, the inferred
values of the parameters are much closer to the ones obtained from the best
fit of the empirical likelihood (dashed red line in both plots), which means
the algorithm is, in this case, pointing in the right direction.

Figure 6: Comparison between the values of the model parameters (left:
$\lambda$; right: $Q$) inferred by the SMC algorithm as a function of time
when directly using the DNS time series (blue curves, time-correlated signal)
and when extracting the detections from the corresponding empirical likelihood
(orange curves, uncorrelated signal). In both cases, each curve corresponds to
the episodes-averaged value of the parameter obtained in one of the $30$
configurations (different source positions) where the algorithm has been
tested. The histograms on the right of each plot have been obtained by
considering the inferred values of the parameters at times $t\geq 300$, while
the horizontal dashed red lines stand for the values of $\lambda$ and $Q$
derived from the best fit of the empirical likelihood. Data shown here
correspond to the setup with $N_{\rm s}=16$ sensors.

Hence, the observed sensitivity of the SMC algorithm in the most realistic
scenario lies in the fact that it must perform a real-time inference of the
model parameters (while looking for the source), which is greatly impacted by
the presence of time correlations in the signal. However, this is not the case
for the WBU approach since it uses a predetermined and discrete set of
parameters, then runs the different models independently, and only at the end
merges their outcome based on the current ranking provided by the overlap
integral of each model. The _static_ nature of the parameter space and the use
of all available (misspecified) models are thus the defining strengths of the
newly introduced approach.

##  IV Discussion

In this work, we revisited the problem of locating an odor source in a
realistic turbulent environment with a network of static sensors. To this end,
we employed DNS of the incompressible 3–D Navier-Stokes equations to simulate
odor dispersion in the atmospherical boundary layer. We then used the data of
the Lagrangian particles trajectories to systematically investigate how the
errors made in modeling such an environment, a problem unavoidable in field
applications, affect the performance of source localization algorithms based
on Bayesian inference.

Within this framework, we identified a quantity that effectively ranks
different models of the environment just based on the history of observations
made by the sensors in the field. This is what we called the _overlap
integral_ $\mathcal{Z}$ as it essentially measures the degree of consensus
among the sensors about a single source location. This quantity, which is
maximized when the model is exact, also represents the posterior estimate of
the probability of making the sequence of observations, integrated over all
possible source locations. Thus, our weighted Bayesian update (WBU) approach
to source localization may be viewed as a form of maximum likelihood
estimation (MLE) applied to the model itself. The use of MLE to quantify the
degree of model misspecification was previously studied in an abstract setting
in Ref. [52].

Through our analysis, we may conclude that WBU is a more robust approach to
locating an odor source in a turbulent environment than state-of-the-art
methods relying on Monte Carlo sampling techniques. In fact, our results
highlight some fundamental weaknesses of the Sequential Monte Carlo (SMC)
algorithm, which features a strong sensitivity to the presence of time/space
correlations in the sensors’ detections, especially in the most general case
where it must also infer the model parameters in real-time together with the
source location. This is in contrast with WBU, which is insensitive to
correlations and proved to be only mildly affected by the use of wrong models.
In particular, WBU, as opposed to SMC, is able to maintain the desired
correlation between the uncertainty about the source location and the distance
of its current estimate from the ground truth. Thus, WBU provides a robust
stop criterion for the search.

The main novelty of the WBU approach stems from the idea that merging the
information gathered from many possible interpretations of the measurements
recorded by the sensors may help in compensating for model errors consistently
with the _many wrongs principle_ [33, 34]. In the realm of olfactory search,
this represents a fundamental stepping stone toward a thorough investigation
of the effect of misspecified models on the localization of odor sources in
turbulent environments, an unavoidable difficulty in practical applications.

The presented results and methodology have clear potential applications in
environmental monitoring [28, 53] and early warnings [29], as they allow to
reliably identify a potential area of intervention when some hazardous
substances are detected. It would also be interesting to explore the
possibility of capitalizing on the use of many wrong models in the case of
olfactory search by single [18, 20] or multiple [19] moving agents, where it
can help mitigate the model misspecification and thus allow for better
decisions of the agent(s). Beyond olfactory search, the WBU approach is, in
principle, also adaptable to any other Bayesian inference problem with model
misspecification. Finally, we suggest that the overlap integral could be
maximized instead by gradient ascent, which would afford a reduction in
computational cost and memory load while losing the extra information from
running many wrong models. It would be interesting, in particular, to explore
the use of a powerful, many-parameter model such as a neural network in such a
context.

## Acknowledgments

We thank M. Sbragaglia and M. Vergassola for useful discussions. We
acknowledge financial support under the National Recovery and Resilience Plan
(NRRP), Mission 4, Component 2, Investment 1.1, Call for tender No. 104
published on 2.2.2022 by the Italian Ministry of University and Research
(MUR), funded by the European Union – NextGenerationEU– Project Title
Equations informed and data-driven approaches for collective optimal search in
complex flows (CO-SEARCH), Contract 202249Z89M. – CUP B53D23003920006 and
E53D23001610006.

This work was also supported by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program (Grant Agreement
No. 882340).

##  Appendix A Analytical argument for the convergence of the overlap integral

Let the true likelihood be $f(h|\bm{r}_{i}-\bm{r})$ and let Bayesian updates
be performed with a model $g(h|\bm{r}_{i}-\bm{r}),$ not necessarily correctly
specified. Let the prior be $\pi(\bm{r}),$ and let the true source location be
$\bm{r}^{*}.$

After $T$ timesteps, we have

$\mathcal{Z}=\int_{\Omega}d\bm{r}\,\pi(\bm{r})\prod_{i=1}^{N_{s}}\prod_{t=1}^{T}g(h_{i}^{(t)}|\bm{r}_{i}-\bm{r})=\int_{\Omega}d\bm{r}\,\pi(\bm{r})\exp\left(\sum_{i=1}^{N_{s}}\sum_{t=1}^{T}\log
g(h_{i}^{(t)}|\bm{r}_{i}-\bm{r})\right).$ (A1)

As long as each observation $h_{i}^{(t)}$ has bounded variance and
correlations between observations decay to zero over time (i.e.
$\mathrm{Cov}(h^{(t)},h^{(t+\tau)})\xrightarrow{\tau\to\infty}0$), we may
apply the (weak) law of large numbers [54], which implies that the argument of
the exponential converges in probability to its expectation over observations
$h$:

$\displaystyle\mathcal{Z}$
$\displaystyle\to\bar{\mathcal{Z}}_{T}\equiv\int_{\Omega}d\bm{r}\,\pi(\bm{r})\exp\left(T\sum_{i=1}^{N_{s}}\sum_{h\in\\{0,1\\}}f(h|\bm{r}_{i}-\bm{r}^{*})\log
g(h|\bm{r}_{i}-\bm{r})\right)$
$\displaystyle=C_{T}\int_{\Omega}d\bm{r}\,\pi(\bm{r})\exp\left[-T\sum_{i=1}^{N_{s}}D_{\rm
KL}(f(\cdot|\bm{r}_{i}-\bm{r}^{*})\parallel
g(\cdot|\bm{r}_{i}-\bm{r}))\right],$ (A2)

where

$C_{T}=\exp\left(-T\sum_{i=1}^{N_{s}}H(f(\cdot|\bm{r}_{i}-\bm{r}^{*}))\right),$
(A3)

$D_{\rm KL}(p\parallel q)$ indicates the Kullback-Leibler divergence between
distributions $p$ and $q,$ and $H$ is the Shannon entropy. By Gibbs’
inequality,

$F(\bm{r})\equiv\sum_{i=1}^{N_{s}}D_{\rm
KL}(f(\cdot|\bm{r}_{i}-\bm{r}^{*})\parallel g(\cdot|\bm{r}_{i}-\bm{r}))\geq
0,$ (A4)

with equality iff $f(h|\bm{r}_{i}-\bm{r}^{*})=g(h|\bm{r}_{i}-\bm{r})$ for each
$1\leq i\leq N_{s}$ and each $0\leq h\leq N_{o}-1$ (where $N_{o}$ is the
number of possible observations, which we have set to two). The set of points
$\bm{r}$ where this holds is the intersection of $N_{s}\times N_{o}$ manifolds
of codimension 1; assuming $f$ and $g$ are well-behaved functions of position,
for large enough $N_{s}$ this intersection should be empty unless
$f(h|\bm{r}_{i}-\bm{r}^{*})=g(h|\bm{r}_{i}-\bm{r}^{*})$ for each $i$ and $h$,
in which case the unique member will be the point $\bm{r}=\bm{r}^{*}.$

Figure A1: (a) Curves of the overlap integral $\mathcal{Z}$ as a function of
the percentage error on either of the model parameters. Different curves refer
to distinct times, as indicated in the legend. (b) Colormap of the overlap
integral $\mathcal{Z}$ at time $t=10^{3}$ as a function of the percentage
error on both model parameters. As expected, it features a maximum when the
model is exact ($\varepsilon_{\lambda}=\varepsilon_{Q}=0$) and decays to zero
as we move away from it. Data shown here have been obtained by averaging over
$10^{2}$ episodes of time length $T=10^{3}$ in units of observations made by
each sensor.

Since $T$ is large, Laplace’s method gives

$\bar{\mathcal{Z}}_{T}\sim
T^{-d/2}C_{T}\sum_{j=1}^{k}a_{j}\exp(-TF(\hat{\bm{r}}_{j}))=T^{-d/2}C_{T}\exp(-T\hat{F})\sum_{j=1}^{k}a_{j},$
(A5)

where $d$ is the number of spatial dimensions, $\hat{F}$ is the minimum of $F$
and is attained on the set $\\{\hat{\bm{r}}_{j}\\}_{j=1}^{k},$ and each
$a_{j}$ is nonnegative and independent of $T.$ Therefore,
$\bar{\mathcal{Z}}_{T}$ is maximized when $f$ and $g$ agree at each
$\bm{r}_{i}-\bm{r}^{*}$ and is exponentially smaller when they do not. This is
the key result.

We can numerically show that the overlap integral $\mathcal{Z}$ relaxes to
zero faster the farther from the ground truth the model is. To this end, let
us take into account the same setup as the one shown in Fig. 2, with the
source located in $\bm{r}_{\rm s}=(92,70)$ and $N_{\rm s}=16$ sensors placed
on a square lattice tilted by an angle $\theta=1.04\,\rm{rad}$ with respect to
the wind direction. We can then use the probabilistic model described in Sec.
II.2 to generate the observations made by the static sensors and set the
source emission rate and the characteristic length scale of the turbulent odor
dispersion to $Q^{*}=5.5$ and $\lambda^{*}=3.5$, respectively. Since we know
the exact model of the detection statistics, we can systematically study how
the overlap integral $\mathcal{Z}$ varies depending on the error made on such
model parameters. Figure A1(a) shows the values of $\mathcal{Z}$ obtained when
either the value of the characteristic length scale $\lambda$ (top panel) or
the one of the emission rate $Q$ (bottom panel) is wrong. In both cases, as
time progresses, the curve of the overlap integral peaks more and more around
the correct value of the parameter, i.e., the one corresponding to a
percentage error $\varepsilon_{*}=0$, and it smoothly goes to zero as the
error grows. More generally, this still holds when both model parameters are
misspecified, as shown in the colormap in Fig. A1(b).

Let us now comment on one feature that stands out from these plots, that is,
the observed asymmetry in the values of $\mathcal{Z}$ when $\lambda$ is
under/over-estimated. In fact, given the same history of measurements, the
source will be estimated as farther from the sensors, the greater the
estimated value of $\lambda$. This will eventually cause the overlap among the
private beliefs to accumulate at the border of the finite-size arena $\Omega$
where simulations are performed. Therefore, the larger values of the overlap
integral $\mathcal{Z}$ when $\lambda$ is overestimated are a mere finite-size
effect. On the one hand, this is not a problem in the case of a functionally
correct model, where the ground truth is attainable (as in this section)
since, eventually, the correct model will still be the one with the largest
value of $\mathcal{Z}$ anyway. On the other hand, however, this effect can
persist in the general case of a functionally wrong model (scenario discussed
in the main text), and we are compelled to address it for consistency. To this
end, during our analysis, we systematically set to zero the values of the
overlap for the models that would place the source at a distance $d\leq
5\Delta x$ (with $\Delta x$ being the lattice spacing) from the border of the
arena $\Omega$. This avoids the selection of clearly wrong models that only
artificially would feature a large value of the overlap $\mathcal{Z}$,
preserving the structure and the basic concepts behind the WBU approach.

##  Appendix B Details on the numerical simulations

In all our numerical simulations, the size of the square arena $\Omega$ has
been set to $128\Delta x\times 128\Delta x$, with $\Delta x=1$ being the
lattice spacing. The sensors have been placed in a square lattice of side
$\ell=88\Delta x$, tilted with an angle $\theta=1.04$ with respect to the mean
wind direction. All the data presented in Sec. III have then been obtained by
averaging over the results obtained by shifting the odor source in $30$
different locations drawn at random within an inner box $108\Delta x\times
108\Delta x$ inside $\Omega$, as illustrated in Fig. A2. This was done to
avoid boundary effects that would have altered the performance statistics of
the localization algorithms.

Moreover, to gather more statistics for each source location, we have divided
the time series of the odor dispersion obtained from the DNS described in Sec.
II.1 into $25$ runs of set time length $T=600$ (in units of the Kolmogorov
timescale $\tau_{\eta}$).

Figure A2: Illustration of the sensors’ placement (magenta circles, here
$N_{\rm s}=16$) and of the $30$ source locations (green squares) drawn at
random inside the arena.

Definition | Symbol | Value
---|---|---
Number of samples | $N$ | 50
Number of MCMC perturbation steps | $K$ | 5
Variance of proposal distribution of $\bm{r}_{\rm s}$ | $\sigma^{2}_{\rm pos}$ | 100
Variance of proposal distribution of $\lambda$ | $\sigma^{2}_{\lambda}$ | 1
Variance of proposal distribution of $Q$ | $\sigma^{2}_{Q}$ | 1
Range of values of $\lambda$ | $[\lambda_{\rm m},\lambda_{\rm M}]$ | [0,10]
Range of values of $Q$ | $[Q_{\rm m},Q_{\rm M}]$ | [0,10]

Table A1: Values of the hyperparameters used in the SMC algorithm.

In order to get the master belief in the WBU approach, we have used in every
configuration a total of $M=100$ models, each of them corresponding to a
different combination of parameters $\\{Q,\lambda\\}$ in the stochastic model
introduced in Sec. II.2, with the specific values of $Q$ and $\lambda$ being
$Q=\\{0.7,1.7,\dots,9.7\\}$ and $\lambda=\\{0.7,1.7,\dots,9.7\\}$,
respectively.

The values of the SMC hyperparameters used in this work are summarized in
Table A1. We have also numerically checked that all the results presented here
are not qualitatively affected by this choice. Note that, in the SMC
implementation, we start from a flat prior defined over the interval $[0;10]$
for both model parameters $Q$ and $\lambda$. Moreover, their values are
bounded therein, which is, for consistency, the same range used in the WBU
approach.

Lastly, the detailed step-by-step explanation of the WBU and SMC
implementation is reported in Algorithm 1 and Algorithm 2, respectively.

Algorithm 1 Weighted Bayesian update

1:for $j=1$ to $M$ do $\triangleright$ Loop over different models

2: Set initial common belief $B^{(0)}_{j}$ to a uniform distribution.

3: Set model parameters $\bm{P}_{j}=\\{Q_{j},\lambda_{j}\\}$.

4: for $t=1$ to $T$ do $\triangleright$ Loop over time

5: Perform sensors’ measurements $\bm{h}^{(t)}$.

6: Update common belief $B^{(t)}_{j}$ using Bayes’ rule (10).

7: Compute overlap integral $\mathcal{Z}^{(t)}_{j}$.

8: end for

9:end for

10:Compute master belief $\mathcal{B}_{M}$ at all times from Eq. (11).
$\triangleright$ Output master belief from weighted models

Algorithm 2 Sequential Monte Carlo with Importance Sampling and perturbation
step

1:Set initial belief $\tilde{B}^{(0)}$ to a uniform distribution.

2:for $t=1$ to $T$ do

3: Perform sensors’ measurements $\bm{h}^{(t)}$.

4: for $i=1$ to $N$ do $\triangleright$ Sample’s initialization

5: Draw sample from current belief:
$\tilde{\bm{\theta}}_{i}\sim\tilde{B}^{(t-1)}(\bm{\theta})$.

6: Compute weight $w_{i}=\mathcal{L}(\bm{h}^{(t)}|\tilde{\bm{\theta}}_{i})$.

7: end for

8: for $i=1$ to $N$ do

9: Normalize weight: $w_{i}/=\sum_{i=1}^{N}w_{i}$.

10: Compute Effective Sample Size: $N_{{\rm eff}}=1/\sum_{i=1}^{N}w_{i}^{2}$.

11: end for

12: if $N_{\text{eff}}<N_{\text{thr}}$ then $\triangleright$ Resampling step
(if needed)

13: for $i=1$ to $N$ do

14: Select $\tilde{\bm{\theta}}_{k}$ with probability $w_{k}$.

15: Put $\tilde{\bm{\Theta}}_{i}=\tilde{\bm{\theta}}_{k}$.

16: end for

17: for $i=1$ to $N$ do

18: Replace $\tilde{\bm{\theta}}_{i}=\tilde{\bm{\Theta}}_{i}$.

19: Set uniform weights: $w_{i}=1/N$.

20: end for

21: end if

22: for $i=1$ to $N$ do $\triangleright$ MCMC perturbation step

23: Select $l\in\\{1,\dots,N\\}$ with probability $w_{l}$.

24: Set $\bm{\theta}^{(t)}_{i,0}=\tilde{\bm{\theta}}^{(t)}_{l}$.

25: for $j=1$ to $K$ do

26: Draw new sample from proposal distribution: $\hat{\bm{\theta}}^{(t)}\sim
q(\hat{\bm{\theta}}^{(t)}|\bm{\theta}^{(t)}_{i,j-1})$.

27: Compute acceptance ratio $\alpha$ from Eq. (12).

28: Draw random number $u\sim U[0,1]$.

29: if $u<\min(1,\alpha)$ then

30: Set $\bm{\theta}^{(t)}_{i,j}=\hat{\bm{\theta}}^{(t)}$.

31: else

32: Set $\bm{\theta}^{(t)}_{i,j}=\bm{\theta}^{(t)}_{i,j-1}$.

33: end if

34: end for

35: Set $\bm{\theta}^{(t)}_{i}=\bm{\theta}^{(t)}_{i,\tilde{B}}$.

36: end for

37: Set
$\tilde{B}^{(t)}(\bm{\theta})=1/N\sum_{i=1}^{N}\delta(\bm{\theta}-\bm{\theta}^{(t)}_{i})$.
$\triangleright$ Output belief approximation

38:end for

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