Model-based Solution Techniques for the Source Localization Problem

Model-based Solution Techniques for the Source Localization Problem Mehmet E. Alpay and Molly H. Shory

Abstract The problem of locating sources in dynamical systems described by partial dierential equations (PDE) is a particular case of the more general class of inverse problems. Source localization problems are often approached using non-model-based techniques. By using a priori knowledge of system dynamics, model-based approaches to this problem can be developed, reducing the number of sensors required to solve the problem. This paper presents three such approaches: o-line numerical computation of the time response data at the sensor(s) from all possible source locations and functions of source strength, spatial and time discretization of the PDE model, and o-line solution of a dual (\forward") PDE problem based on the adjoint system model. In each case, a particular algorithm is presented, and analysis of appropriate sensor placement and the minimal number of sensors required is given. In all three approaches, a minimal amount of on-line processing is required. The relative strengths and shortcomings of the three approaches are discussed and are demonstrated through application to the two-dimensional isotropic heat equation. Keywords - partial dierential equations, point sources, inverse problems, sensor placement, number of sensors, solution algorithms.

1 Introduction A by-product of the technological advances during the twentieth century has been the everincreasing accumulation of chemical and radioactive wastes throughout the world. Such materials usually take quite a long time to decompose and consequently have to be stored in special storage facilities until they can be safely disposed [1], [2]. The risks inherent in long-term storage of dangerous substances include human exposure during monitoring and routine maintenance, danger to both humans and the environment from  This research is supported in part by NSF Grants 9309336, 9312745-ECS and Sandia National Laboratory Contract LF7363 (Principal investigators are Alexander Khapalov, Dept. of Mathematics, Washington State University, and Ronald Mohler, Dept. of Electrical and Computer Engineering, Oregon State University) y Department of Electrical and Computer Engineering, Oregon State University, Corvallis, OR 97331-3211, Email: [email protected], [email protected]

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undetected leaks, and human exposure during clean up of detected leaks. The longer it takes to detect and contain a leak, the worse the damage to the environment and the greater the cost and the risk of human exposure required for the subsequent clean up. Monitoring may be automated by placing sensors and instrumentation within the storage facilities to detect the occurrence of leaks and to nd automatically the source of any leaks. The advantages of this approach are that

 human presence is not required for routine monitoring;  leaks may be detected and their locations found promptly, allowing immediate containment or repair, which minimizes the amount of spillage and the cost of the subsequent clean up;

 the human exposure required to nd the source of the problem and to contain it is minimized. The main disadvantage is that maintenance and calibration are required for the sensors and instrumentation, which may result in some level of human exposure. To minimize the exposure required for routine maintenance and calibration, the number of sensors should be minimized. Some redundancy may be necessary to ensure reliability. To implement an eective monitoring system, one needs a strategy for the placement of the monitoring sensors and an algorithm to evaluate the data collected by these sensors. We identify the following key issues:

 On how to select the number and location of the sensors:

{ The number of sensors should be sucient to detect the occurrence of a leak and to nd the source of the leak. 2

{ The number of sensors should be minimal to minimize human exposure during routine maintenance and calibration.

 On what algorithm(s) to use to evaluate the data coming from the sensors:

{ Such algorithms should be capable of automatically evaluating the sensor data and deciding whether or not a leak has occurred, and, if it has, where the leak source is.

{ The on-line computations must be fast and accurate to be of any practical use. The physical phenomenon of a leak in a storage area for hazardous materials can be modeled as a distributed parameter system described by partial dierential equations since the quantity of interest, and which could be measured, is a continuous function of space as well as time [3]. In this paper, we study such systems and try to address the issues listed above. In particular, we develop three algorithms for using point-sensor data to locate point-sources and propose strategies for placing the sensors. In each case, we assume that there is a single source concentrated at a single point driving the system, and that measurements are taken at pre-determined points in the region of interest - hence the terms point-source and point-sensor. For the rst two algorithms, the source strength is assumed known. The rst algorithm, the intelligent brute-force method, is based on the simulation of every possible scenario and comparison of the actual measurement data with the source signatures collected from these simulations. The recommended sensor locations for this method are those locations where the minimum dierence between the various source signatures is maximum. This choice decreases the sensitivity of the algorithm to measurement noise and model mismatches. This approach requires prior knowledge of the source strength variation in time, which is the main limiting factor in its application. The second algorithm, the discretization method, involves the spatial discretization of the partial 3

dierential equation governing the system in space, which yields a lumped linear time-invariant state-space model. This state-space model is used to obtain an input-output model describing the relationship between the source and the measurements. The input-output model is then discretized in time. The resulting relationship is used to construct an algorithm that identi es the source location for a given sequence of measurement data. This method can handle cases where the source strength is not known, although it does assume that the unknown source strength remains constant throughout the measurement period. The third method, the dual system method, involves the construction of a system of partial dierential equations dual to the original one and computation of this new system's steadystate response to a source of constant strength. (A source in the dual system corresponds to a sensor in the original system.) The duality relationship between the two systems, the precomputed steady-state response, and the measurement data from the actual system are then used to identify the location of the source driving the actual system. This method can handle sources of unknown time-varying strength. It also provides a means of qualitatively assessing sensor positions and choosing where to place the sensors to achieve a reliable identi cation of the source location. It should be noted at this point that the problem we address in this paper is an inverse problem (see [4] for a general treatment of such problems). Inverse problems are notorious for their ill-posed nature; i.e. small perturbations in the data used for solving such problems may yield

signi cantly dierent results (see, for example, [5]). Various solution algorithms for dierent inverse problems and their convergence and accuracy properties have been extensively studied in the literature (see [6], for example, for a literature survey of the studies in the inverse radiative transfer problems). For a given system, there are many possible ways of posing an inverse problem. Inverse problems 4

can be classi ed in two general groups:

 Given the system characteristics, nd the input yielding a measured output.  Given the input and output, nd the system (parameters) yielding the observed inputoutput pair (system identi cation). Most studies of inverse problems deal with problems of the second type ([7] and [8], for example) and they are focused on determining the properties of the medium within which the process takes place. Our study is fundamentally dierent from such work since we assume that we have a priori knowledge of the system model and characteristics, and we try to solve an inverse problem of the rst type. There are also a number of studies involving the development of solution algorithms to the source localization problem for particular cases. Examples of such work can be found in the eld of acoustics (see [9], [10] and [11], for example), nuclear engineering ([12]), etc. [9] and [10] are based on the concept of spatial ltering. The aim of spatial ltering is to arrange a set of sensors in such a way that signals that have overlapping frequency content but originate from dierent spatial locations are separated. In [11], on the other hand, the researchers use a sensor array to estimate the angles and relative arrival times of various multipath trajectories from the source. These trajectories are then back-propagated in time and space to obtain an estimate of the source location. In an earlier study ([12]) on the detection and localization of radioactive sources within an environment, researchers consider a transport process in a slab bounded by two parallel planes. They assume that the radiation inside this medium is due to continuously-distributed (in space) internal isotropic sources, and that the radiation is isotropically absorbed and scattered within the medium. To estimate the spatial source distribution from experimental measurements, the 5

researchers approximate the system of PDE's for the angular ux by a system of ODE's. This is followed by the application of quasi-linearization to the system of ODE's too reconstruct the source distribution yielding the measured outputs. The studies cited above rely heavily on the characteristics of the particular medium within which the source localization problem is to be solved. As a result, the solution techniques developed in these studies are mostly case-speci c and might not be suitable for other source localization problems. Our approach diers from such work in that we develop general approaches that can be tailored to various systems of interest. Our results apply to many systems for which deterministic mathematical models with known parameters can be developed. In the following sections, we provide a more precise description of the type of systems to which our approaches apply and de ne the source localization problem for such systems. We then develop the three algorithms mentioned above and illustrate how they work by applying them to the two-dimensional isotropic heat equation. We conclude this paper with a discussion of the relative merits and shortcomings of each method.

2 System Model and Problem Formulation As mentioned earlier, a leak in a storage area can often be modeled by the changes it induces on the distribution of some physical quantity (e.g., temperature, radioactivity level, concentration of a chemical in air/water/earth) in the storage area. Such distributions are typically governed by partial dierential equations, which model distributed parameter systems. One example is the two-dimensional isotropic heat equation describing the temperature distribution in an area:

@u(x; y; t) = c @ 2 u(x; y; t) + c @ 2 u(x; y; t) + f (x; y; t): @t @x2 @y2 6

This equation models the distribution of temperature u(x; y; t) in time and space in response to the heat source f (x; y; t). The term isotropic refers to the fact that the environment within which heat diuses is a uniform one, which is why the terms @ 2 =@x2 and @ 2 =@y2 have the same diusion constant c. Another example is the two-dimensional neutron transport equation, which models the neutron ux in response to a radioactive source. A two-dimensional version of such an equation, discretized in angle, can be given as follows: 1 @ m (x; y; t) +  @ m (x; y; t) +  @ m (x; y; t) + s (x; y; t) = m m  @t @x @y 4 m M q(x; y; t) ; m = 1; 2; : : : ; M: t X ! ( x; y; t ) + k k 4 k=1 4 In the equation above, m is the total neutron ux in the direction speci ed by index m, m and m are the neutron mobility constants in directions x and y respectively, and t is the scattering constant. The weighted sum on the right-hand side of the equation represents the total neutron ux. The term q(x; y; t) represents the source driving the system. De ning a ux vector = [ 1 2


M ]T and appropriately de ning matrices Ax , Ay , B and Q, the above set of equations can be rewritten as

@ (x; y; t) = A @ (x; y; t) + A @ (x; y; t) + B (x; y; t) +  Q(x; y; t) x y @t @x @y 4 The two systems described above are special cases of the general system model

@u(~r; t) = Au(~r; t) + F (~r; t) @t

(1)

where ~r is the position vector, F (~r; t) is the source driving the system and A is an operator including the space dierentiations and ordinary multiplications of the quantity u. For a system of the general form (1), we assume that the source term F (~r; t) is of the particular form

F (~r; t) = v(t)(~r ? ~rso ): 7

(2)

The source is an impulse in space with possibly a time-varying source strength function v(t). The problem of interest is to nd the spatial location ~rso of the source. De ne as the spatial set within which the problem is to be solved and @ as the boundary of this region. We assume that

u(~r; 0) = 0 for all ~r 2 ;

u(~r; t) = 0 for all t when ~r 2 @ ;

(3)

i.e., there is no initial distribution and the boundary distribution is zero at all times. (At present, our method of constructing the adjoint operator requires us to assume a given distribution on the boundary of the spatial region.) The system output, measured at a nite number of point-sensor locations, can be expressed as an output vector

h

y(t) = u(~rse1 ; t) u(~rse2 ; t)


u(~rseN ; t)

i

(4)

where ~rsei denotes the location of the ith point sensor. Our goal is to develop methods that use the above sensor data to determine the source location

~rso. For each method we devise, we attempt to provide techniques for evaluating potential sensor locations so that not only a minimum number of sensors are used but also the source location can be uniquely and reliably determined. Ideally we would like to have full continuous-time data for each sensor measurement, but we recognize the impracticality of this and implement our algorithms using the time-sampled sequence of the sensor data instead. We develop our algorithms for the general system and output formats stated in (1), (2), and (4), and illustrate their performance via computer simulations on the two-dimensional isotropic heat equation. For the intelligent brute force method, we assume that the source strength v(t) is a known function. For the discretization method, we assume it is an unknown constant. For the dual system method, the source strength may be an unknown time-varying function. The method performance is better, however, for certain types of source strength functions - those 8

that do not grow exponentially fast.

3 The Intelligent Brute Force Method For this rst method, we assume that the possible sources are in some known set - with possible sources at certain known locations and the source strength a known function of time. We also assume that numerical simulations can be performed o-line on a computer model of the system for all possible sources. We de ne the signature of a particular source at a sensor location as the response measured by a sensor at that location when that particular source is activated. (A unique signature for a given source will be obtained at a sensor location only if the source strength is a known function.) We can compare the signatures for dierent sources at a sensor location using a signal norm. We de ne the signature dierentiation capacity (s.d.c.) of a sensor location as the minimum distance between any two source signatures at that sensor location, over the set of all pairs of sources. In mathematical terms s.d.c.(x) = min jjs (x) ? sj (x)jj i6=j i

(5)

where si (x) and sj (x) are the signatures of the sources i and j with respect to position x, and

jj
jj is an appropriate signal norm. The intelligent brute force method requires the computation of the signature dierentiation capacities at all possible sensor locations. The sensor(s) should be placed at location(s) with the highest signature dierentiation capacity. The signatures at the selected sensor locations of all possible sources are then stored in a memory bank, for on-line comparison with the actual measurements from the sensor(s). The on-line computations are the distances between the measurements and the stored signatures, using the signal norm that was used to select sensor 9

locations. The smallest distance(s) select the likely source location. In mathematical terms, Source = arg min jjsi(
) ? y(
)jj i where jj
jj is the same signal norm used to evaluate the signature dierentiation capacities of sensor locations. The o-line computations proposed are extensive, but need be done only once. They are used to optimize the placement of the sensors, and to bank the signatures associated with given source locations. The number of o-line computations increases as O(n2 ) with the number n of possible source locations. The on-line computations are reasonable, and scale up linearly O(n) with the number of possible source locations. The intelligence of this method is that it provides a sensible criterion for selecting sensor locations, resulting in signatures that are distinct enough to dierentiate between the likely sources. The signatures are functions of time, so a relatively small number of sensors are likely to be required, as compared to using only steady-state information for the dierentiation purposes. (Of course, it is necessary to keep such a system running at all times so that it will pick up the signature of an event when it happens.) It is simple to decide how many sensors are required to dierentiate between possible sources. If there is a single sensor location for which all source signatures are suciently dierent from one another, than a single sensor at such a location is enough for source location identi cation purposes. If not, then a set of appropriate sensor locations can be selected as follows:

Theorem 1 Let sij (
) be the signature of source i with respect to sensor location j , i = 1; : : : ; N , j = 1; : : : ; M . Let Se = fj1 ; : : : ; jm g be a subset of possible sensor locations. Then sensors located at positions depicted by set Se can identify any source location if and only if for all

1  i1 ; i2  N , there exists jk 2 Se such that jjsi1 jk (
) ? si2 jk (
)jj 6= 0, i.e. given any source 10

pair, there exists at least one sensor that can distinguish their signatures.

4 The Discretization Method 4.1 Introduction The discretization method is based on the discretization of the original partial dierential equation describing the system, successively in space and in time, to obtain a simple inputoutput relation at the sensor locations. This input-output relation is then used to evaluate the sensor data and to pick the likely source location. A step-by-step summary of the procedure is given below: 1. Partition the space using a uniform grid. (This may be generalized.) 2. Discretize the space derivatives in the partial dierential equations describing the system using the partition obtained in the previous step, and obtain a lumped ordinary dierential equation state-space model for the system. (This may be done by identifying an ODE model that matches the data well at the steady-state values of the system, for the possible sources of interest.) 3. Decouple the state-space equations to obtain an input-output model at the sensor location. 4. Discretize the input-output model in time for analysis purposes and obtain a simple autoregressive (AR) model whose parameters depend on the source location in a prescribed manner. 5. Analyze the available output data to identify the source location using the nal autoregressive model developed in the previous step. The mathematical derivations for this method are in the following subsection. 11

4.2 Mathematical derivations This mathematical details for this procedure are as follows. 1. Let us partition the region using a uniform grid and let ~r1 ; : : : ;~rn be the corresponding grid points. De ne

h

u = u1 u2


u n

ui (t) := u(~ri ; t);

iT

Let
be the area of a grid cell. The source, which is a spatial impulse in the original problem, is approximated by a pulse located at (unknown) ~r` with the same (unknown) volume  as the spatial impulse. 2. The spatial derivatives included in operator A of (1) are approximated by weighted differences to obtain the system matrix G. The sensors are located at or near grid points

~rm1 and ~rm2 . We thus obtain the following state-space model ( rst-order matrix ordinary dierential equation) approximation

du = Gu +  b " dt# " T #
y1 = c 1 u y2 cT2

(6)

where b and ci , i = 1; 2, are column vectors with

(

6= ` bj = 10 ifif jj = `

(

if j = mi cij = 01 otherwise In equation (6),  denotes the unknown (constant) strength of the source driving the system. The factor (1=
) comes as a result of approximating a spatial impulse of unit volume with a spatial pulse of base area
and height 1=
. As a result of this approximation, a source term is present in the simpli ed system equations only for the state variable ul 12

that is within one cell size (
) neighborhood of the source location. Consequently, the source vector b (6) has zeros at all locations but one. 3. Now let us assume that the characteristic equation for the matrix G is given by

Gn + p1Gn?1 +


+ pn?1 G + pnI = 0

(7)

Then the state-space equations in (6) can be decoupled at each sensor locations to obtain the input-output models: T n?1 n?2 yi(n) + p1 yi(n?1) +


+ pnyi =  (1 | =
)ci (G + p1{zG +


+ pn?1I }) b i = 1; 2: (8) qiT

4. Using backward Euler discretization to discretize (8) in time, we obtain the following auto-regressive model:

0yi;k + 1 yi;k?1 +


+ nyi;k?n =  qiT b i = 1; 2: where the coecients j , j = 0; :::; n are given by

j

= (?1)j

nX ?j

m=0
t

1

n?m

(9)

!

n?m p m j

where p0 = 1 and
t is the time increment. 5. The right-hand side of (8) is a scalar constant ( times the `th element of qiT ), for both

i = 1; 2. Since we wish to determine `, we would like to compute the left-hand side numerically from the sensor data to get an approximate value, and then select the element of the vector qiT closest to the value. However, since  is an unknown constant, we must modify this idea slightly. We evaluate the left-hand side on-line, recursively, for each sensor, using fy1;k g and fy2;k g. The ratio, + 1 y1;k?1 +


+ n y1;k?n k = 0 yy1;k + y +


+  y 0 2;k

1 2;k?1

13

n 2;k?n

(10)

of the left-hand sides, is equal to q1T b=q2T b = q1;` =q2;` . De ning

q = q1 :=q2 as the constant vector formed by element-by-element division of q1 by q2 , one can then choose the entry of the vector q closest to the ratio k , for k large. The index of this entry of the vector (`) gives our best estimate of the source location ~^rso , to be the middle

p

of the cell denoted by `, i.e., at ~r` . This will be correct to a precision of
n=2, where

n is the dimension of the physical space in which the problem is set. The inner workings of this method are exempli ed in Section 6.2, where the method is applied to the two-dimensional isotropic heat equation. It should be noted that while the constant source strength restriction for this method is a severe one, this method is still superior to the brute force method in the sense that it can handle sources with unknown strengths, provided that they are known a priori to remain constant throughout the period of observation.

5 The Dual System Approach 5.1 Introduction This method involves the construction of a system model dual to the original system (1), (2), (4) and o-line computation of the steady-state response of the dual system to particular inputs (sensors). The steady-state response of the dual system, together with the duality relationship between the two systems, is then used to solve for the unknown source location. The idea of constructing the dual system and employing the duality relationship between the primal and dual systems as a means of solving this problem was suggested to us by Alexander Khapalov [13]. We have improved on his approach by introducing the geometry of the steadystate solution to the dual system as a problem parameter, which enabled us to devise our 14

solution algorithm. The mathematical derivations for this method are given in the following subsection.

5.2 Mathematical derivations 5.2.1 Dual system The following system is dual to the system model described by (1), (2), and (4):

@z(~r; t) = ?A z(~r; t) + (t)(~r ? ~r ) se @t

(11)

where

t 2 T = (0; ); ~r 2