ON SOURCE-TERM PARAMETER ESTIMATION FOR LINEARADVECTION-DIFFUSION EQUATIONS WITH UNCERTAIN

COEFFICIENTS

SERGIY ZHUK∗, TIGRAN TCHRAKIAN∗ , STEPHEN MOORE† , RODRIGO

ORDONEZ-HURTADO‡ , AND ROBERT SHORTEN∗

Abstract. In this paper, we propose an algorithm estimating parameters of a source term ofa linear advection-diffusion equation with an uncertain advection-velocity field. First, we apply aminimax state estimation technique order to reduce uncertainty introduced by the coefficients. Thenwe design a source localization algoritm which uses the state estimator as a model and estimatesthe parameters of the source term given incomplete and noisy data. The principal novelty of theproposed algorithm is in that it is robust with respect to the uncertainty in advection coefficients,i.e. wind fields. The localization algorithm is sequential, that is it updates both state estimateand source estimate once a new observation arrives. To demonstrate the efficacy of the proposedalgorithm, we present a numerical example of source localization in two spatial dimensions for theadvection-dominated transport of a non-reactive pollutant emanating from a point-source.

Key words. advection diffusion equations, PDEs, state estimation, minimax, source localiza-tion.

AMS subject classifications. 35K20, 65L60, 93E10

1. Introduction. Generic Advection-Diffusion-Reaction (ADR) equations arewidely used to model and forecast a wide range of atmospheric processes: a good ex-ample is given by forecasting ozone concentrations at ground level [11]. An importantclass of ADR equations is represented by linear Advection-Diffusion (AD) equationswhich can be used to describe a variety of processes involving heat or mass transfer inwhich chemically reactive effects are absent or negligible. Examples include the evo-lution in concentration of a chemically non-reactive pollutant in a fluid, or the changein heat of a non-reactive flowing substance. In this paper, we study the problem ofestimating the parameters of source-term included in an AD equation as a means ofdescribing an exogenous input. Many of the previous works in this area have been inthe context of air pollution [13]. This is a suitable application of advection-diffusionmodels, since many pollutants of interest can be assumed to be non-reactive withthe atmosphere. Furthermore, the point-source emission of such a pollutant (e.g. anatural gas leak) can be modelled using a source term. The parameter estimationproblem could then be to determine the parameters of the source term describing theemission given measurements. For example, the location, intensity and time of originof the point source may be estimated using measurements provided by the sensorslocated at points in the spatial domain.

In this work, the advection coefficients of the AD equation is taken to be uncertain.Specifically, we assume that a nominal time-independent advection field, U is givenbut the advection filed of the signal, observed through the sensors, ~U(x, t) is time-

dependent, that is ~U(x, t) = U + V (x, t) where V (x, t) is a bounded divergence-free perturbation. In the context of a gas leak problem, this means that while thespeed and direction of the prevailing wind are known to be represented by U , its

∗IBM Research - Ireland, Damastown, Dublin 15, Ireland(sergiy.zhuk,robshort,tigran@ie.ibm.com)†IBM Research, Melbourne, Australia(steve.moore@au.ibm.com)‡Maynooth University, Maynooth, Co. Kildare, Ireland, and University College Dublin, Belfield,

Dublin 4, Ireland (rodrigo.ordonez-hurtado@ucd.ie)

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actual direction ~U(x, t) is uncertain and changes over time. This uncertainty in theadvection velocity field is a key differentiating feature of our problem and, in thecontext of a point-source emission model, plays an important role in the transport ofa pollutant. In many previous studies concerning the emission estimation problem,wind is assumed to be available and can be treated in a feed-forward manner. Clearly,in many practical situations this assumption is unrealistic. In this paper we developa new data fusion algorithm for source-detection and localization that is robust withrespect to wind uncertainty.

A suitable application of our problem is the estimation of parameters describinga point-source emission into the atmosphere. Thus, we consider the typically en-countered scenario of a gas leak in an urban setting: a gas (for example, methane)plume with intensity I originating at coordinates (x∗) begins at time t∗. We assumethat the gas is either non-reactive1 or nearly inert so that the motion of the plumecan be modelled reasonably well by a linear advection-diffusion equation (as detailedin Section 2.1). The model begins by assuming zero concentration. An abrupt gasemission is modelled by a source term represented as a product of Gaussian functions(centered at x∗ with standard deviation correlated with grid size (see section 2.1) anda Heaviside function s (switching from 0 to I at time t∗). In fact, this source termmodels the continuous emission of the gas at constant rate and fixed location. Wewill refer to this example throughout the paper.

The model is then projected onto a finite dimensional subspace generated by La-grange polynomials by using Spectral Element Method (SEM) and we arrive2 at anOrdinary Differential Equation (ODE) describing the dynamics of projection coeffi-cients. In what follows we refer this ODE as SEM model. Our choice of SEM wasmotivated by the convenience it affords in dealing with boundary conditions3 andexpedient sparsity patterns for mass and stiffness matrices. In addition, the solutionof AD equation is at least 2 times differentiable (in the weak sense) almost every-where in the domain of interest Ω, provided Ω has a “nice” boundary. Thus, we mayuse high-order SEM approximations to attain better convergence (over elements) andmaintain non-negativity of the solution almost everywhere in Ω. Note that gas flow isadvection dominated (diffusion magnitude is proportional to 10−6) and so various nu-merical instabilities may arise. These instabilities are usually overcome by introducingnumerical diffusion in specific directions depending on the advection coefficients [12].To simplify the presentation this paper uses the standard SEM method and the nu-merical stabilisation is achieved by a combination of high order Lagrange polynomialsand uniform numerical diffusion.

Note that the SEM model is subject to the following uncertainty sources: (i)uncertain but bounded source term (or input) quantifying projection error, and (ii)model error. In the context of the gas leak example, the latter accounts for uncertaintyin the wind field. Specifically, the stiffness matrix of SEM model is a function of theadvection coefficients describing the velocity field. In its interpretation as a wind-field,the advection velocity is usually approximated by a weather model which provides anominal (or mean) component and bounds (or confidence intervals) for possible windfluctuations. Hence, the SEM model has a nominal stiffness matrix and a model error

1A non-reactive gas is a gas that does not undergo chemical reactions under a set of givenconditions.

2For convenience of the reader a brief derivation of this ODE is given in the appendix.3Indeed, it is shown in the appendix that little or no work is required to implement a weak form

of mixed homogeneous Neumann/non-homogeneous Dirichlet boundary conditions

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accounting for these fluctuations (see sections 2.2-2.3). Following [25] we suggest totake into account both projection and model errors by introducing a minimax filter.We refer the reader to [14, 2, 10, 15, 23, 24] for the basic information on minimax stateestimation framework. The minimax filter constructs a linear estimate of the SEMmodel’s state vector given observations, provided the uncertain parameters (projectionerror, model error and observation error) belong to a given bounding set. Statistically,the latter assumption implies uniform distributions for uncertain parameters and,under these assumptions, the minimax filter is designed so that for any realisation ofuncertain parameters the estimation error is minimal. Being quite conservative, theaforementioned uncertainty description (in terms of a priori bounding set) is alignedwith the requirements of our application as reliable statistics from the wind fields, orfrom projection errors, are often unavailable. As a result, our model of gas transportis represented by the minimax filter. This is a linear ODE which is composed of threeterms:

• a non-stationary state transition matrix representing an estimated nominalwind field;

• a source term represented by a linear transformation of observations;• and finally, a source term s(t, t∗)L(x∗) where L represents the spectral el-

ement projection of the product of Gaussians modelling the point source,which depends on uncertain emission coordinates x∗, and s is a Heavisidefunction switching from 0 to I at uncertain time t∗.

The problem of source localization is then recast as the following optimisation prob-lem: minimize (w.r.t. x∗, I and t∗) a data misfit function, d(Y −HC) measuring thedistance between a vector of observed concentrations at given sensor’s locations, Yand the corresponding components of the minimax filter’s state vector, HC. In otherwords, we define source location and emission time so that the vector of concentrationsestimated by the minimax filter is as close as possible (in terms of a chosen data misfitfunction d) to the observed concentrations. Note that the state vector of the filter isa linear transformation of the source term sL. Thus, by introducing a set of adjointODEs we may recast the aforementioned optimisation problem as a non-linear regres-sion problem: minimize d(Y − ΛsL) w.r.t. x∗ ∈ Ω, 0 < I < I∗ and 0 < t∗ < T wherethe matrix Λ does not depend on x∗ and is computed beforehand by using solutions ofadjoint ODEs (see section 3). This optimisation problem is non-convex and we employa sampling strategy to find its solution. There are many techniques for dealing withnon-convex optimization problems. For convenience, here we apply a particle swarmmethod to perform a sequential localization, i.e. the estimate of uncertain x∗, t∗ andI at time instant t = tk is updated once a new observation Y (tk+1) arrives. We de-tail this computational strategy and the corresponding numerical results in sections 3and 4.

As we have already mentioned, the principal novelty in our approach is to usethe minimax filter as a model of transport (i.e. gas transport). The filter constructsa set, E which contains the “vector of true concentrations” C∗, provided the uncer-tain parameters belong to the given bounding set. In fact, the minimax center ofE, C coincides with the estimate of the concentrations provided by the filter. Thisparametrisation of the problem provides a formalisation of intuitively obvious relationbetween the quality of source localization and sensor topology. Namely, by adjusting(x∗, I, t∗) we steer the minimax center of E, the state of the minimax filter towardsobserved concentrations (tracking). This procedure by itself does not give us anymeasure of localization quality. However, we know that C∗ ∈ E and so, by varying

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sensor topology we can adjust the volume of E: if the volume of E is getting smallerover time then the “vector of true concentrations” C∗ becomes closer to the center ofE, C and that, in turn improves source localization quality. This feature of the min-imax filter affords a systematic degree of freedom to the designer of a sensor networknot normally available. Specifically, by appropriately dimensioning and positioningthe sensor network the designer may thus explore the trade-off between the predictionaccuracy and network cost. This basic intuition is formalised in section 3.

Brief overview of related literature. The use of Kalman filter for gas leak local-ization purposes is a common direction in the literature (see [20, 21] and referencestherein). The drawbacks associated with Kalman filters are well documented. Theseinclude: unrealistic assumptions on noise; sensitivity to errors in initial estimates ofcovariance matrices [18]. A further complication in our context is the assumption,made by many authors, that the source can be represented as a linear combination ofbasis functions, and that these coefficients do not vary over time. The consequence ofthis is that the dimension of the state vector grows rapidly and the problem quicklybecomes intractable [20]. In our case the source is effectively parametrised by 4 param-eters (location, intensity and emission time) and so the main computational burdenis to compute the minimax filter. The latter may be computed without any specialtreatment in two dimensions. In 3D one may use localised minimax filters obtainedby means of a domain decomposition strategy [16]. A further point to note in thecontext of Kalman filters is that the model error accounting for the wind uncertaintyis unlikely to be drawn from normal distribution4. In contrast, we assume that themodel error is merely bounded. Another typical assumption is that the given coef-ficients may indeed reproduce the “true concentration”. In other words, the wind isusually assumed to be known which is rarely the case in practice.

A parametrisation of the source as a Dirac distribution was used in the context ofsource localization by various authors. For instance, Bayesian source localization al-gorithms locating point sources for simple Gaussian puff models, which admit an ana-lytic solution, were proposed in [5]. A branch and bound based alternative to standardnon-linear least squares localization for the model and source of the same type wassuggested in [13]. A stochastic gradient method was applied in [17] for the advection-diffusion equation with stationary wind, homogeneous Dirichlet boundary conditionsand point source. In contrast, our source model uses a weak approximation of theDirac delta in the class of smooth functions in order to achieve numerical advantages(see section 2.1). Generic source localization algorithms for full advection-diffusionequations with mixed type boundary conditions were considered in [1]. Source lo-calization by using adjoint solutions of advection-diffusion equations in 3D togetherwith uncertainty quantification algorithm has been proposed in [22]. The setup of thelatter paper is quite close to our setting. However, as we have already mentioned,the key differentiator is that we allow for uncertain advection coefficients. Control-lability and identifiability conditions of the source for advection-diffusion equationswere derived in [9]. In our context these conditions cannot be applied due to the timevarying nature of our problem. Finally, balancing uncertainty of the source localiza-tion and energy costs of sensor’s network load was considered in [20] in the Kalmanfiltering framework and this is related to the trade off between number of sensors, andlocalization accuracy, that we consider briefly in section 4.

4As it depends on the state which is, in turn, assumed to be driven by “white noise” and so themodel error becomes a “coloured noise”.

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2. Problem statement. Our approach is based on three components: first, weintroduce a linear advection-diffusion equation and discretise it in space using SpectralElement Method (SEM); second, we apply a minimax filtering algorithm to accountfor uncertain advection coefficients and also for numerical projection errors. Thirdand finally, we derive a dual form of the localization problem which is of low dimensionand hence may be efficiently solved by sampling. In what follows we briefly describethese components.

2.1. Advection-diffusion equations. To simplify the presentation we do allthe following derivations in two dimensions. We stress that the derivations below canbe easily generalized to the case of n dimensions. Consider a linear advection-diffusionequation of the following type:

(2.1) ∂tu+∇ · (~U(x, t)u)−∇ · (ε(x)∇u) = s(t; t∗)g(x− xs) .

Here u(x, t) stands for the concentration (i.e. gas concentration) at the point x =

(x1, x2)> and time instant t, ~U = (v1, v2)T is the given wind field, the term ∇ ·(~U(x, t)u) describes how the concentration u is transported by the wind ~U and ∇ ·(ε(x)∇u) defines the rate of the dissipation of u in Ω.

The shape of the source is given by the function, g. We model the source by whatis called a “nascent delta function”, that is a smooth function which converges to thedelta function in the sense of distributions. Specifically,

(2.2) g(x− xs) = Cε−1e−(x1−x

∗)2−(x2−y∗)2

ε2 ,

where xs = (x∗, y∗)> is the source location, C is a normalisation constant and ε > 0is correlated with the spatial discretisation as detailed in section A.1.

We further assume that the initial concentration is zero, u(x, 0) = 0, and afterthe gas has been emitted at time 0 < t∗ and location xs ∈ Ω, its transport is governedby (2.1) subject to the following mixed boundary conditions:

u(x, t) = 0 for x ∈ Γ1 anddu

dn(x, t) = 0, for x ∈ Γ2 ,

where dudn denotes the derivative of u in the direction n, n is the unit normal vector

of Γ1, where Γ1 ∪ Γ2 = ∂Ω, ∂Ω denotes the boundary of Ω, and Γ1 ∩ Γ2 = ∅. Thefirst condition simply means that nothing enters the domain through Γ1 and theconcentration may leave the domain through Γ2 according to the second condition.

Finally, the position of the source, xs within Ω is assumed to be uncertain togetherwith emission time t∗ ∈ (0, T ), and s(t; t∗) is a function which activates the source,namely s(t; t∗) := IH[t− t∗] where H[·] is the Heaviside step function and 0 ≤ t∗ ≤ T .The intensity of the emission is given by I and we assume that it does not change overtime. In addition, we suppose that the wind field ~U may be uncertain as is usuallythe case in practice. Specifically, we assume that ~U(x, t) = U(x) + V (x, t) where Uis a nominal stationary wind field and V (x, t) is bounded and describes fluctuations

of the measured/estimated/predicted wind field ~U around U . We assume that U andV (x, t) are divergence free, that is div(U) = 0 and div(V ) = 0. The latter assumptionimplies that the advection field preserves the volume of the concentration within thedomain.

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2.2. SEM model. To construct a discrete representation of the physical model (2.1)we apply the Spectral Element Method (see appendix A) and arrive at the followingdiscrete in space but continuous in time representation of (2.1):

(2.3)dC

dt= A(U)C +A(V (t))C + s(t; t∗)L(xs) , C(0) = 0 ,

where the state transition matrix, A(U) ∈ Rn×n encapsulates the wind field compo-nent U and the diffusion component, and matrix A(V (t)) corresponds to the windfield component V . Note that A(U) is a sum of skew-symmetric matrix A0 (repre-senting divergence-free advection field U) and a symmetric negative definite matrixA1 (representing diffusion), and A(V (t)) is a skew-symmetric matrix (representingdivergence-free perturbation V ). The vector L(xs) ∈ Rn represents the source posi-tion in the chosen set of the basis functions. Since the basis is formed using Lagrangeinterpolation polynomials (see Appendix A), L(xs) is simply a grid-function approxi-mating g(x−xs) at the computational nodes (GLL points) as discussed in section A.1.

We stress that the norm of C is not increasing over time provided sL = 0. Indeed,if we multiply (2.3) by C> we obtain by skew-symmetry of A0 and A(V ) that:

1

2

dC>C

dt= C>(A(U) +A(V (t)))C = C>A1C ≤ 0 .

This observation motivates our uncertainty description (2.5) in the following section.

2.3. Minimax filtering. In order to account for uncertain advection coefficientsand also for numerical projection errors, and to incorporate data from the sensors intothe SEM model we apply the minimax filtering. To this end, we assume that a functionY (t) ∈ Rp is observed and

(2.4) Y (t) = HC(t) + η(t) ,

where η is noise and H is an observation matrix relating the synthetic vector ofconcentrations, C(t), to the observed concentrations Y (t) ∈ Rp. In fact, C(t) repre-sents concentrations computed by the SEM model at the computational nodes (GLLpoints). For simplicity we assume here that the sensors are located at computationalnodes and in this case the matrix H takes the most simple form, namely H is theidentity matrix if there are gas measurements at all GLL points. In the case where pis less that the number of GLL points, the matrix H comprises only the correspond-ing p rows of the full H (identity matrix). It is not hard to generalise this setupto the case of arbitrary sensor locations within the computational domain by simplysetting H to be an interpolating matrix5. Using this convention the entries of thevector Y describe the observed concentrations from the network of sensors located atcomputational nodes.

We define e(t) := A(V (t))C to be a model error and suppose that

(2.5)

∫ T

0

e>(t)Q(t)e(t) + η>(t)R(t)η(t)dt ≤ 1

where Q(t) = Q>(t) > 0 and R(t) = R>(t) > 0. Here Q and R are weighting matricesassociated with the model and set of sensors. Specifically, matrix Q defines the L2-magnitude of the deviation of the error caused by the wind fluctuation e(t) from the

5The matrix of values of Lagrange polynomials over the desired sensor’s locations

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nominal wind field. This information should be provided by the weather model orobtained experimentally. The matrix R defines the magnitude of the observation errorη and characterises the reliability of the sensors. In what follows we assume that thesematrices are given.

Finally, given this basic setup we would like to estimate the position of the source,namely (x∗, y∗) ∈ Ω, emission time 0 < t∗ < T and source intensity 0 < I < I∗.

3. Source localization. For a fixed s(t; t∗)L(xs) we define the minimax filterestimating the vector of synthetic concentrations C(t) as follows:

(3.1)dC

dt= A(U)C + P (t)H>R(t)(Y (t)−HC) + s(t; t∗)L(x) , C(0) = 0 ,

where P solves the differential Riccati equation, i.e.

(3.2)dP

dt= A(U)P + PA>(U) +Q−1(t)− PH>R(t)HP , P (0) = 0 .

Let us also define β2 as a solution of the following ODE:

β2 = (Y (t)−HC)>R(t)(Y (t)−HC) , β2(0) = 0 .

It then follows that for all 0 < t ≤ T we have:

(3.3) (C(t)− C(t))>P−1(t)(C(t)− C(t)) ≤ 1− β2(t)

where t 7→ C(t) verifies (2.3)-(2.4) for some e, η satisfying (2.5) and the latter inequal-ity holds true for any such C. In other words, the above ellipsoid represents the setof all reachable states of (2.3) compatible (or conditioned) on data Y .

The above interpretation suggests to use the minimax filter C as a transportmodel. Indeed, note that

• the filter constructs the ellipsoid (3.3), E which contains the “vector of trueconcentrations” C∗ corresponding to the “true” source vector sL, providedthe uncertain parameters belong to the given bounding set;

• the minimax or “Tchebyshev” center of the set E defined by (3.3) coincideswith the estimate of the concentrations provided by the filter.

Therefore we may reformulate the localization problem in terms of the minimax filter:adjust the source parameters (x∗, y∗, t∗) so that the minimax filter C becomes as closeas possible (in some norm) to the vector of the observed concentrations. Since C isthe center of E and the truth belongs to E, it then follows that by adjusting thesource sL we are steering the entire ellipsoid E towards “the true ellipsoid” E∗. Westress that the volume of the ellipsoid reflects the amount of the uncertainty whichis accumulated in the system due to model and observation errors. In particular, Eshrinks if Q and R are large enough reflecting the situation of little or no model andobservation errors. On the other hand, the source vector which minimizes the distancebetween C and Y is robust with respect to the observation and model errors.

Recall from section A.1 of the appendix that L(xs) := M−1Fr (s − KFxCFx) and

s = Φq where Φ is a known matrix and q is a grid function representing g (see (2.2)).We can therefore split (3.1) into independent equations:

dC1

dt= A(U)C1 + P (t)H>R(t)(Y (t)−HC1)−M−1

FrKFxCFx , C1(0) = 0 ,

dC2

dt= (A(U)− P (t)H>R(t)H)C2 + s(t; t∗)M−1

FrΦq , C2(0) = 0 ,

(3.4)

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so that C = C1 + C2 and C1 depends only on the data and is independent of thesource.

Let us define the following cost function,

(3.5) J(x∗, y∗, I, t∗) :=

Nt∑k=1

d(Y (kh)−H(C1(kh) + C2(kh))) ,

where h := TNt

and d(·) is a norm (either 2-norm or 1-norm).Let us now define the adjoint variable λnj as follows:

(3.6)dλkjdt

= −(A(U)− P (t)H>R(t)H)>λkj , λkj (kh) = Hj ,

where Hj is the j-th column of H>. Integrating by parts it is easy to get that

H>j C2(kh) = I

∫ kh

t∗(M−1

FrΦ)>λkj (t)>dtq) .

Let us define a matrix Λ := ∫ kht∗

(M−1FrΦ)>λkj

p,Ntj,k=1 and set Y := Y (kh)−HC1(kh)Ntk=1.

Then

J(x∗, y∗, I, t∗) := d(Y − Λ(t∗)Iq) .

Clearly, J is non-convex and non-smooth (in case of d representing 1-norm) functionof (x∗, y∗, I, t∗). In the following section we employ a sampling strategy to find aminimizer of J . The key computational bottleneck of the sampling is in computingΛ. Since each column of this matrix may be computed independently it follows thatan appropriate distributed computing strategy may be applied to compute Λ ratherquickly.

3.1. Particle Swarm Optimizers. Particle Swarm Optimization (PSO) is aheuristic optimisation approach which has been inspired by the social behaviour ofbiological organisms [8, 3]. In PSO, a “swarm” of M candidate solutions p(i) isevolved. Here, let p(i) := (x∗i , y

∗i , t∗i , Ii)

> denote the ith candidate for the solutionof J → minΩ×[0,T ]×[0,I∗]. Each particle p(i), i = 1, . . . ,M is updated through twoequations of motion of the type

(3.7) v(i)k+1 =

ωkv(i)k +

(gk − p(i)

k

)+ ρk (1− 2r2,k) , if i = iτ ,

ωkv(i)k + φ1

(l(i)k − p

(i)k

)+ φ2

(gk − p(i)

k

), otherwise.

,

(3.8) p(i)k+1 = p

(i)k + v

(i)k+1 ,

φ1 = c1r1,k, φ2 = c2r2,k ,

l(i)k = argmin

1≤j≤kJ(p

(i)j ),

gk = argmin1≤i≤M,1≤j≤k,

J(p(i)j ),

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where ωk is a time-variant inertia weight, c1,2 are acceleration coefficients, r1,k andr2,k are independent random vectors with uniform distribution in (0, 1), iτ is the index

of the best particle so p(iτ )k = gk, and ρk is a time variant scaling factor defined as

(3.9) ρk+1 =

2ρk, if #successesk > sc,0.5ρk, if #failuresk > fc and ρk > εm,ρk, otherwise,

where εm is the smallest allowed value for ρk given by machine precision, #successesand #failures are the number of consecutive successes/failures respectively, a successis defined as J (gk) < J (gk−1), and a failure is defined as J (gk) = J (gk−1). In orderto ensure that (3.9) is well-defined, it is also necessary to use the following rules:

#successesk+1 > #successesk

⇒ #failuresk+1 = 0,

#failuresk+1 > #failuresk

⇒ #successesk+1 = 0.

For more details on the implementation of the PSO used in this paper we referthe reader to [19].

4. Numerical results. In this section, we describe the numerical results ofthe source-localization. In place of a physical experiment, we use the SEM-discretisedadvection-diffusion model to generate observations (measurements) which we are thenused for the localization, which is also based on the SEM model. In order to generatemore realistic data, the SEM model used for the observations uses two wind compo-nents as described in Section 2.2, each one a velocity field. The nominal velocity-field,U(x) describes the speed and direction of the prevailing wind. This stationary com-ponent is assumed to be known, and thus appears in the localization algorithm. Theother wind field, V (x, t) is oscillatory (non-stationary) and is present in order to em-ulate the randomness of the wind. This component is not assumed to be known, andis used only for the generation of observations. In other words, the observations aregenerated by the SEM-discretisation of the advection-diffusion model with the truesource, and with state transition matrix, A = A(U) +A(V ), while the localization isperformed using the SEM model using only A(U). Note that A corresponds to bothan advective and diffusive component.

Discrete advection-diffusion equation. The advection-diffusion equation (2.1) isdiscretised in space using 18 × 18 square elements and 4th order Lagrange poly-nomials over the spatial domain Ω = [0, 1]2. The resulting grid is represented onFig. 4.2b. We impose homogeneous Dirichlet boundary condition over the bottomand left sides of Ω and homogeneous Neumann conditions over top and right sides.The SEM model (2.3) is then discretised in time over [0, T ] with T = 2 and time steph = 0.01 by using implicit mid-point method as detailed in appendix B. The nominalvelocity-field, A(U) is presented in Fig. 4.1 (red arrows) and the oscillatory compo-nent A := τA1 + (1 − τ)A2 where A1,2 are represented by left and right (w.r.t. thered arrow) black arrows and τ is drawn at each time step from a uniform distributionsupported over [0, 1]. Note that typically a gas flow is advection dominated (diffusionmagnitude is proportional to 10−6) and so various numerical instabilities may arise.These instabilities are usually overcome by introducing numerical diffusion in specific

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x1

x2

Fig. 4.1: Wind-field over a single element

directions depending on the advection coefficients [12]. To simplify the presentationwe use uniform numerical diffusion (ε(x) = 0.01). We run the simulated physicalexperiment with source placed at x∗ = 0.2333, y∗ = 0.15 and starting at t∗ = 0.1000,and the emission intensity is I0 = 0.09. The solution of SEM model (2.3) correspond-ing to A = A(U) + A(V (t)) and “true” source, i.e. source of intensity I0 placed at(x∗, y∗) and starting at t∗, is presented in Fig. 4.2. In Fig. 4.2a, the leak has alreadystarted but it has not reached the sensors yet. We then see its spread across thedomain in Figs. 4.2b–4.2f. Clearly, the gas leaves the domain through the top andright boundaries and is 0 at the bottom and left boundaries of the domain so thatthe boundary conditions are verified. Note that, the dynamics of the concentration isadvection driven and is quite sensitive to wind fluctuations (see Fig. 4.2). The Figs4.2c–4.2f also show localization results which will be discussed below.

Minimax filter. To perform the localization we generated 2 different sensor’s lay-outs randomly. In Figure 4.3, we show 6 sensors representing the 1st layout, and thesimulated data “observed” at each sensor location. The sensor readings were taken atevery time step so that in total each sensor generated 2

h = 200 readings. Figure 4.3ashows 3 utmost left sensors in the 1st layout, and the concentration observed by thosesensors is shown in Figure 4.3c. Figures 4.3b and 4.3d show the 3 utmost right sensorsof the 1st layout and the associated data. In Figure 4.3c, we see that the gas reachesthe sensors at different times, based on their respective distances from the source.The concentration seen on a sensor depends on the wind direction and the distancefrom the source. We can see that the sensors are quite far from the actual location ofthe leak and hence the plume reaches sensors only after 60 timesteps. In addition, itcan be observed that concentration observed by 3 utmost right sensors (see Fig. 4.3d)increases and decays over time due to the changing wind direction. This shows thatthe popular constant wind Gaussian puff models does not apply in this situation.

Both wind uncertainty and number of observations affect the minimax estimate,C, which in turn affects the localization. In Figure 4.4, the true concentration iscompared versus the minimax filter for different numbers of observations at a singlequadrature point. Note that the filter is computed given the “true” source to illus-trate the impact of the wind uncertainty and sensor locations. Filter parameters are

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time = 0.3 Sensors = 6 Wind = 1 Intensity = 0.97786 (0.09) Emission time = 0.1 (0.1)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(a) Simulated gas leak at t = 0.15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time = 0.7 Sensors = 6 Wind = 1 Intensity = 1 (0.09) Emission time = 0.1 (0.1)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(b) Simulated gas leak at t = 0.30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time = 1 Sensors = 6 Wind = 1 Intensity = 0.12191 (0.09) Emission time = 0.1 (0.1)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(c) Simulated gas leak at t = 0.70

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time = 1.5 Sensors = 6 Wind = 1 Intensity = 0.088414 (0.09) Emission time = 0.1 (0.1)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(d) Simulated gas leak at t = 1.00

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time = 1.8 Sensors = 6 Wind = 1 Intensity = 0.094221 (0.09) Emission time = 0.1 (0.1)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(e) Simulated gas leak at t = 1.30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time = 2 Sensors = 6 Wind = 1 Intensity = 0.092669 (0.09) Emission time = 0.1 (0.1)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(f) Simulated gas leak at t = 1.40

Fig. 4.2: Advection-diffusion of a point source over time

taken to be R = 100 (we trust the sensor readings) and Q = 0.001 (large model erroris allowed to compensate for the wind), S = 1 (initial concentration fluctuates aroundzero to accomodate a possible source emission close to the initial time step). Intu-itively, more observations are expected to result in a better estimate. An example ofthis is seen upon comparing Figures 4.4a and 4.4b, from which we see that the mini-

11

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x1

x2

0

0.5

1

1.5

2

2.5

3

(a) Grid and 3 utmost left sensors

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x1

x2

0

0.5

1

1.5

2

2.5

3

(b) Grid and 3 utmost right sensors

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time

Polluta

nt concentr

ation

Sensor A

Sensor B

Sensor C

Source start−time

(c) Observed data from 3 utmost left sensors(in fig above)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time

Polluta

nt concentr

ation

Sensor A

Sensor B

Sensor C

Source start−time

(d) Observed data from 3 utmost right sensors(in fig above)

Fig. 4.3: Sensor positioning on the computational grid (top row) and associated ob-servations (bottom row)

max estimate of the concentration using 26 sensors is the best. However, this is notalways the case; a small number of well-placed sensors may provide more informativeobservations than a larger number of sensors.

Localization. Finally, we apply PSO to approximate the global minima of J . Forthe experiment we take d as the L1-norm, i.e. the sum of absolute values. Notethat the localization algorithm uses the following information: if the leak is firstdetected at time t = ts, then the search for the start-time t of the leak should beperformed on the interval, [0, ts]. Figures 4.3 show different ts for different sensor’sconfigurations. To localize in space, find emission time and intensity we evaluated the

matrix Λ(s, k) = ∫ khsh

(M−1FrΦ)>λkj

p,Ntj,k=1 for Hj running through all columns of H>

provided s ranges from 0 to Ns where Ns is chosen so that Nsh = ts, and k variesfrom Ns + 1 up to Ns + 150. The latter shows that we used just 150 sensor readingsout of available 200.

We initialize the search as follows: set i = 1, get the vector of observations,Y (ts + ih), use a swarm of M randomly generated particles p(i)Mi=1 with M = 50,

12

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time

Polluta

nt concentr

ation

Number of observations = 6

True concentration

Estimated concentration

(a) 6 sensors

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time

Polluta

nt concentr

ation

Number of observations = 26

True concentration

Estimated concentration

(b) 26 sensors

Fig. 4.4: True and estimated concentration for different numbers of sensors at a singlequadrature point

and do 200 iterations of the equations (3.7) and (3.8), i.e. for k = 1, . . . , 200. Thenwe find the best particle out of M particles, g200,i. We then employ the followingprocedure: we use Y (ts + ih), Y (ts + (i+ 1)h) and generate (randomly) a swarm ofM − 2 particles which is then appended by g200,i and a particle which represents theposition, detection time and intensity provided by the sensor which has the earliestactivation time. Once the best particle is found, say g200,i+1, we extend the setof observations by Y (ts + (i + 2)h) and repeat the aforementioned procedure untilY (ts + 150h) is included in the set of observations. We use for PSO the followingvalues for the configuration parameters: ωk decreasing linearly from 0.5 down to 0.2,c1,2 = 2, sc = 15 and fc = 5.

Fig. 4.5a shows the evolution of the source location estimate over time: we cansee that y-component converges to 0.1605 which is quite close to the true locationy∗ = 0.15 and x-component converges to 0.1642. The uncertainty in the estimate isdue to (i) highly oscillating wind (see Fig. 4.2), (ii) sensor positions which are quitefar off the actual leak location (see Fig. 4.3), and (iii) the averaging effect of theminimax filter (see Fig. 4.4) which is required to mitigate the uncertainty brought bythe fluctuating wind. Visually, the localization results in space (estimate at a giventime step is marked by the white cross) can be assessed by looking upon Figs. 4.2d-4.2f.At the same time, the source emission time and intensity are identified almost exactly(see Figs 4.5b-4.5c). Fig. 4.5d shows the dynamics of the relative L1-estimation error(for all the components of g200,t = (x∗t , y

∗t , t∗t , It)

>) which stabilizes at aronund 14%.Both layouts (6 and 26 sensors) exhibit similar behaviour even though the ramp-uptime of the second layout is a little longer. Estimation results for both layouts arepresented in the Table 4.1 below.

5. Conclusion. We developed a framework for gas leak source-localization usingan advection-diffusion equation with uncertain coefficients describing the transport ofa point-source pollutant. We used the minimax state estimator as a robust model ofgas transport and designed a localization procedure by minimising a distance betweenthe output of the estimator and the data. The results of synthetic experiments arepromising. The proposed localization procedure may process observations online. An

13

Time0 50 100 150

Pos

ition

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PSO estimate of xPSO estimate of yTrue position x = 0.23True position y = 0.15

(a) Source location (x∗t , y∗t )

Time0 50 100 150

Em

issi

on ti

me

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

True emission time t=0.1PSO estimate

(b) Source emission time t∗

Time0 50 100 150

Inte

nsity

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PSO estimateTrue intensity I=0.09

(c) Source intensity I0

Time0 50 100 150

Rel

ativ

e L1

err

or

0

0.5

1

1.5

2

2.5

3

3.5

6 sensors16 sensors

(d) Relative L1 estimation error

Fig. 4.5: PSO estimations given by g200,t = (x∗t , y∗t , t∗t , It)

> for the increasing numberof observations t = 1 : 150 (layout of 6 sensors)

Layout (x∗, y∗, t∗, I) Error in x∗ Error in y∗ Error in t∗ Error in I6 (0.1642, 0.1605, 0.1000, 0.0926) 0.0691 0.0105 0.0000 0.002626 (0.1519, 0.1472, 0.1000, 0.0936) 0.0815 0.0028 0.0000 0.0036

Table 4.1: Source estimates and localization error (True source: (0.2333, 0.1500,0.1000, 0.0900))

interesting reseach direction would be to define a minimal number of sensors and theirpositions to guarantee a prescribed quality of the localization and consider severalsources. We leave this as a direction for further research.

A. Spectral element method formulation for 2D advection-diffusionequation. The SEM is based upon the classical Galerkin projection procedure [6],that is to look for solutions of a PDE in the space, spanφn(x1)φm(x2)n,m=1...N+1.

14

The concentration can then be approximated by u, which is the truncated series,

(A.1) u(x1, x2, t) =

N+1∑n,m=1

unm(t)φn(x1)φm(x2) =

N+1∑n,m=1

unm(t)φnm(x1, x2),

where unm are projection coefficients. In SEM, basis functions φm are chosen so thatthe coefficients unm approximate the values of the solution u(x1, x2, t) at a specifiedset of points on Ω (as detailed below). Replacing u in (2.1) with its approximation,u, we obtain the residual,

RN+1 :=

N+1∑n,m=1

unmφnm + v1

N+1∑n,m=1

unm∂x1φnm + v2

N+1∑n,m=1

unm∂x2φnm

− ε1

N+1∑n,m=1

unm∂x1x1φnm − ε2

N+1∑n,m=1

unm∂x2x2φnm − g(x1, x2, t)

=

N+1∑n,m=1

(unmφnm +

2∑p=1

vpunm∂xpφnm −2∑q=1

εqunm∂xqxqφnm

)− s(t; t∗)g(x, t)

(A.2)

where the dot denotes time-differentiation. In order to incorporate mixed Neu-mann/Dirichlet boundary conditions into the weak formulation of the problem werequire the projection of the residual onto Ψ0 := spanφij , φij = 0 on Γ1 to vanish,i.e.,

(A.3) 〈RN+1, φij〉Ω = 0, φij ∈ Ψ0 ,

where 〈a, b〉Ω =∫Ω

abdΩ. Consequently, (A.3) reads as follows:

N+1∑n,m=1

∫Ω

unmφnmφijdΩ +

2∑p=1

∫Ω

vpunm∂xpφnmφijdΩ−2∑q=1

∫Ω

εqunm∂xqxqφnmφijdΩ

= s(t; t∗)

∫Ω

g(x, t)φijdΩ, i, j = 1 . . . N + 1.

(A.4)

Using integration by parts on the third integral on the left hand side of (A.4) givesus the following weak formulation of the equation (2.1):

N+1∑n,m=1

∫Ω

unmφnmφijdΩ +

2∑p=1

∫Ω

vpunm∂xpφnmφijdΩ +

2∑q=1

∫Ω

εqunm∂xqφnm∂xqφijdΩ

= s(t; t∗)

∫Ω

g(x, t)φijdΩ, φij ∈ Ψ0,

(A.5)

where we applied (i) homogenous Neumann boundary conditions, and (ii) the defi-nition of Φ0. The above weak formulation is now used as a starting point for SEM.

15

The latter involves dividing the spatial domain into a set of non-overlapping elements,

Ω =Ne⋃e=1

Ωe and constructing basis functions φeij compactly supported over the ele-

ments. Specifically, over each element Ωe we define basis functions as follows:• φeij(x1, x2) = `Ni (ξe1(x1))`Nj (ξe2(x2)) for (x1, x2) ∈ Ωe,• φeij(x1, x2) = 0 for (x1, x2) 6∈ Ωe if φeij(x1, x2) = 0 on the boundary of Ωe,

where (ξ1(x1), ξ2(x2))> maps Ωe into a reference element Ωc := [−1, 1]2 and `Ni areLagrange polynomials

(A.6) `Ni (ξ) =

N+1∏j=1,j 6=i

ξ − ξjξi − ξj

i = 1, 2 . . . , N + 1

with ξj ∈ [−1, 1] denoting the Gauss-Legendre-Lobato (GLL) quadrature points.Note that `Ni (±1) = 0 if i = 1 or i = N + 1 and `Ni (±1) = 1 otherwise and soφe1j , φ

eN+1j , φ

ei1, φ

eiN+1 6= 0 on ∂Ωe. Let us take any of these functions, say φe1j .

Then there exist an adjacent element Ωa(e) and a basis function φa(e)jN+1 such that

φa(e)jN+1 = φe1j on the interface between Ωe and Ωa(e). Note that φ

a(e)jN+1 = 0 (φe1j = 0)

on the boundary of Ωa(e) (Ωe), which is opposite to the aforementioned interface.

Thus we can extend6 φe1j to Ωa(e) setting φe1j = φa(e)jN+1 over Ωa(e). Using the same

argument we can extend φe1j over A(Ωe), the set of all elements adjacent to Ωe, and

set it to be 0 outside A(Ωe). As a result we obtain an H1-basis function with com-pact support over A(Ωe). Clearly, φeij ∈ Ψ0 provided A(Ωe) does not intersect with

Γ1. Taking this into account we set u(x1, x2, t) =∑Ne,N+1e,n,m=1 unm(t)φen,m(x1, x2) and

rewrite (A.5) as follows:

N+1∑n,m=1

∫A(Ωe)

unmφenmφ

eijdΩ +

2∑p=1

∫A(Ωe)

vpunm∂xpφenmφ

eijdΩ +

2∑q=1

∫A(Ωe)

εqunm∂xqφenm∂xqφ

eijdΩ

= s(t; t∗)

∫A(Ωe)

g(x, t)φeijdΩ, φeij ∈ Ψ0 ,

(A.7)

This intuitive description is made precise in [7, p.40, p.49, p.56].Using Gauss-Legendre-Lobato quadrature to compute integrals in (A.7) we arrive

at the following system of ODEs:

(A.8) MFrCFr +KFrCFr +KFxCFx = s(t; t∗)s ,

where the components of the mass matrixMFr are given byMenm,ij =

∫A(Ωe)

φenmφeijdΩ,

the components of the stiffness matrixKFr are given byKenm,ij =

∑2p=1

∫A(Ωe)

vp∂xpφenmφ

eijdΩ+∑2

q=1

∫A(Ωe)

εq∂xqφenm∂xqφ

eijdΩ. Here the basis functions φemn belong to Φ0 as well as

test functions φeij so that MFr and KFr are square matrices. Note that MFr is di-

agonal due to the cardinal property of the Lagrange polynomials, `Ni (ξj) = δij . The

6This extension is continuous with L2-derivatives and so it is of H1-class that agrees with thefact that equation (2.1) is parabolic and has weak solutions of class H1.

16

matrix KFx is obtained in a same way as KFr but with basis functions φemn 6∈ Φ0

so that KFr is rectangular. The vector of concentrations, C = (CFr, CFx)> approxi-mates nodal values of u that is the actual values of the concentration u over the setof nodes given by GLL points7 Here CFr represents the nodal values of u which arenot defined by Dirichlet boundary conditions. These values correspond to the basisfunctions from Φ0. CFx represents nodal values over Γ1 which are given by Dirichletboundary conditions and are therefore fixed. These values correspond to φemn 6∈ Φ0.Finally, s is a projection of the source term (as detailed in the following section) ontoΦ0 with components sij =

∫A(Ωe)

g(x, t)φeijdΩ.

We rewrite (A.8) as

(A.9)dC

dt= AC + s(t; t∗)L(xs) ,

where A = −M−1FrKFr and L(xs) := M−1

Fr (s−KFxCFx).

A.1. Source term. We now consider the source term in more detail. Since theflow is advection dominated it follows that implementing the source as a delta-functionmay induce oscillations rendering the scheme unstable. For this reason, we model thesource by what is called a “nascent delta function”, that is a smooth function whichconverges to the delta function in the sense of distributions. A possible choice of such

a function is g(x− xs) = ε−1Ce−(x1−x

∗)2−(x2−y∗)2

ε2 where xs = (x∗, y∗)> is the source

location, C−1 :=∫

Ωe−(x1−x∗)2−(x2−y∗)2

dx1dx2 is a normalisation constant and ε > 0.In order to evaluate sij =

∫A(Ωe)

g(x, t)φeijdΩ we note that ε > 0 needs to be

correlated with the spatial discretization. To illustrate this on a simple exampleconsider 1D case:

ε−1

∫ L1

0

e−(x−x∗)2

ε2 φi(x)dx ≈ ε−1Nx∑n=0

e−(xn−x∗)2

ε2 φi(xn)∆x

where ∆x = L1

Nxand xn := n∆x. If we now set ε := ∆x we obtain:

N∑n=0

e−(xn−x∗)2

∆x2 φi(xn) (∗) .

Note that the values of the exponential function in the above sum may be consideredas weights assigned to the values of φi over the uniform grid given by points xn.Clearly, for our choice of ε = ∆x the exponential vanishes very rapidly away fromx∗, provided Nx is sufficiently large. As a result, only a few adjacent grid points xnhave actual impact (numerically) on (∗). If x∗ = xn for some n it then follows thate· = 1 and φp(x

∗) receives the highest weight and so it has the largest impact onthe value of (∗). Hence, for N → ∞ the above sum converges to φp(x

∗). The sameholds true for the 2D case as basis functions and g are separable. As a result, theprojected source term is represented by s = Φq where Φ is the matrix such that itse i j row is given by the values of the basis function φeij ∈ Φ0 over the uniform gridon Ω with step ∆x = ∆y and q is a grid function representing g over the same grid.

7This follows from the fact that we used a nodal expansion for u so that u(x1, x2, t) =∑Ne,N+1e,n,m=1 unm(t)φen,m(x1, x2) is actually an interpolation of a function with values umn(t) at (m,n)-

node of the grid represented by GLL points.

17

B. Numerical solution of Ricatti equations. In order to solve (3.4) and (3.2)we employ the method proposed in [4]. Let us introduce a uniform grid on [0, T ] withstep h > 0 and define Uj and Vj from the following system of equations:

(B.1)( I+h

2A> h

2H>RH

h2Q−1 I−h2A

)( Uj+1

Vj+1

)=( I−h2A> h

2H>RH

h2Q−1 I+h

2A

)(IPj

),

where Pj := VjU−1j for j > 0 and P0 = 0. We also define and Cj solves the following

system of linear equations:

(I − h

2A+

h

2Pj+1H

>RH)Cj+1 = (I +h

2A− h

2PjH

>RH)Cj +h

2

Pj+1H>RYj+1 + PjH

>RYj2

,

(B.2)

where C0 = 0. The discretization for C1 is not required. Instead, to compute λkj(see (3.6)) one uses exactly the same discretization as shown in (B.2) but backward intime. Since the scheme (B.2) is symmetric it does not change when one reverses thetime: namely, to integrate forward in time one uses initial value of C0; to go backwardone starts from the final value and uses the same scheme.

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