Computational Studies on Detecting a Diffusing Target in a Square Region by a Stationary or Moving Searcher

In this paper, we compute the non-detection probability of a randomly moving target by a stationary or moving searcher in a square search region. We find that when the searcher is stationary, the decay rate of the non-detection probability achieves the maximum value when the searcher is fixed at the center of the square search region; when both the searcher and the target diffuse with significant diffusion coefficients, the decay rate of the non-detection probability only depends on the sum of the diffusion coefficients of the target and searcher. When the searcher moves along prescribed deterministic tracks, our study shows that the fastest decay of the non-detection probability is achieved when the searcher scans horizontally and vertically.


Introduction
Search problems arise commonly in many diverse areas [1].For instance, we look for a missing key or person, the police officers search for fugitives, and prospectors explore for mineral deposits.Systematic research on search problems is now commonly known as search theory, which traces its root to the need of detecting surfaced U-boats either visually from aircraft or with radar during World War II [2]- [7].
In search theory, the object sought is called the target.The problems can be loosely divided into three categories: a stationary target encountering a moving searcher, a moving target encountering a stationary searcher, and a moving target encountering a moving searcher.Much of the literature prior to the 1970s focuses on stationary targets.A comprehensive survey on research literature on moving targets has been provided by Benkoski et al. [8].
In [9], Eagle considered the problem of a stationary searcher looking for a single moving target.He obtained an analytical expression for the non-detection probability of a randomly moving target encountering a stationary sensor when the search region was a disk and the cookie-cutter detector was fixed at the center of the search region.Mangel [10] [11] looked at the problem where a target was assumed to move in the plane and the searcher in space.Optimal search path problems have been addressed by Washburn [12] [13], Eagle and his co-workers [14]- [17].The conflict between simplicity and optimality in searching for a 2-D stationary target was dealt with by Washburn [18].A sequential approach to detect static targets with imperfect sensors such as tower-mounted cameras and satellites was presented by Wilson et al. [19].Majumdar and Bray derived the survival probability of a tracer particle moving along a straight line in the presence of diffusing traps in the plane [20].Fernando and Sritharan calculated the non-detection probability of infinitely many diffusing Brownian targets by a moving searcher which travels along a deterministic path with constant speed in the two-dimensional plane [21].In this paper, we compute the non-detection probability of a diffusing Brownian target in the presence of a stationary or moving searcher in a square region.We study the effect of sweeping paths by considering five scenarios: the search may 1) diffuse randomly, 2) move along a circular or square loop, 3) move along a spiral, 4) move along a square spiral, and 5) scan horizontally and vertically.

Problem Setup
Consider a square region with half width A R , centered at the origin in the two-dimensional space.Mathemati- cally, the square can be described as [ ] [ ] , , . In our search problem, this square is the search region in which the target undergoes Brownian diffusion.
Suppose the searcher is capable of detecting a target instantly when the target gets within distance R to the location of the searcher and there is no possiblity of detection when the target range is greater than R.That is, the searcher covers a disk of radius R centered at the location of the searcher.The expression "cookie-cutter detection rule" is often used to describe this type of sensor modeling.One major criticisim of the cookie-cutter rule is based on the argument that fluctuations in the performance of detection equipment and human operators make it extremely rare to have a critical detection range R. Despite the limitations, the cookie-cutter model offers the simplest and most practical method to model sensors including radar, eyeball, infra-red, and low level TV.We illustrate the search problem in Figure 1.We carry out Monte Carlo simulations to study the time evolution of the non-detection probability, respectively, when the searcher is fixed at various locations, when the searcher undergoes Brownian diffusion with various values of diffusion coefficient, and when the searcher is moving along various prescribed deterministic paths.
Let t D denote the diffusion coefficient of the target, and s D the diffusion coefficient of the searcher.In our simulations, we choose the parameters as follows: and consider five problems below.

Problem 1: Diffusing Target and Diffusing Searcher
We look at the situation where the target and the searcher are diffusing with various diffusion coefficients.The case of a stationary searcher is the special case with 0 , is the location of the target at time , is the location of the searcher at time .
x j y j t x j y j t

= ∆
In Monte Carlo simulations, we advance the target and the searcher in time according to where 1 W , 2 W , 3 W , and 4 W are independent samples of standard normal distribution (mean 0, variance 1).To enforce the reflection condition at the boundary of the square search region, we calculate the new positions of target and searcher as where function

( )
Reflection x is defined as The target is labeled as "detected" at j t if the distance between the target and the searcher is less than the detection range R: .
Once the target is detected, that particular Monte Carlo run is terminated and another independent Monte Carlo run is started.To speed up the simulation, multiple Monte Carlo runs are carried out in parallel.
Let ( ) 0 0 , x y be the initial location of the searcher.In Monte Carlo simulations, the initial location of the target is selected randomly and uniformly from the part of the square search region that is outside the disk of radius R centered at ( ) 0 0 , x y (i.e., outside the the searcher's detection area at 0 t = ).For each set of parameter values, we repeat the Monte Carlo run 100000 N = times.The non-detection probability is calculated by averaging over 100,000 repeats.
In Problem 1, we select the time step That is, the root-mean-square of the displacement between the target and the searcher in time period ∆ is no more than one eighth of the detection radius of the searcher.
We first examine the accuracy of our Monte Carlo simulations in the case of ( ) ( )     Recall that the ini- tial location of the target is randomized and is most likely away from the center.0 t D = fixes the target at its initial off-center location and makes it less likely for the diffusing searcher to encounter the target.In the case of 0 t D = , if we switch the roles of the target and the searcher, we see that when a searcher is fixed at an off-center location with no diffusion, the non-detection probability decays the slowest.Thus, for a given relative diffusion between the searcher and the target ( ) 4 suggests the following observations: 1) the decay rate of the non-detection probability is the largest when the searcher is fixed at the center; 2) when both the searcher and the target are diffusing with significant diffusion coefficients, the decay rate of the non-detection probability is lower and is independent of s D as long as s t D D + is a fixed constant; 3) when the searcher is fixed at a location significantly off center, the decay rate of the non-detection probability is even lower.
To further test these observations, we compare the decay rates of the non-detection probability for 4 sets of parameters Based on the observations 1)-3) above, we expect that set 1 produces the fastest decay of the non-detection probability; sets 2 and 3 yield similar decay rates, lower than that of set 1; and set 4 gives the slowest decay rate of the non-detection probabilty.
Figure 5 compares the results for these four parameter sets.The results in Figure 5 confirm what we predicted based on observations 1)-3).Hence, these results provide further support for observations 1)-3).Figure 5 also indicates that when the searcher has significant diffusion, its inital location does not matter.
Next we study the case of a fixed searcher ( ) We investigate how the searcher's location affects the decay rate of the non-detection probability.Figure 6 shows that for a stationary searcher, the decay rate of the non-detection probability decreases as the distance between the searcher and the center is increased.
In summary, for Problem 1, we conclude that a) when both the searcher and the target have significant diffusion, the decay rate of non-detection probability is independent of the initial location and is independent of s D as long as s t D D + is fixed; b) for a given value of s t D D + , the fastest decay of non-detection probability oc- curs when the searcher is fixed at the center ( ) 0, 0 .Next, we study the case where the searcher moves with a constant velocity s v along a prescribed determinis- tic loop.

Problem 2: Searcher Moving along a Loop
We consider the situation where the target diffuses with diffusion coefficient 100 t D = and the searcher moves with velocity s v along a loop (a circular or a square loop).We select velocity s v as follows.Let 1 τ be the time scale of the target diffusing a root-mean-square distance of 2R along a given direction.

Time scale 1
τ can be viewed as the time scale of the target probability distribution relaxing to erase the mark swept by the searcher.Time scale 1 τ is given by ( ) The distance traveled by the searcher with velocity s v in time period 1 τ is 1 s v τ .We consider the regime  where the target velocity is neither too small nor too large.Specifically, we consider the case where the distance traveled by the searcher in time 1 τ is a small multiple of 2R: ( ) We pick 10 α = .The corresponding velocity is found to be In all the simulations below, we use 500 In Problem 2, for each set of parameter values, we repeat the Monte Carlo run 200000 N = times.We also use a smaller time step (see below).The increased time and ensemble resolution is made possible by the fact that when the searcher moves along a loop, the non-detection probability decays faster than in the optimal case of Problem 1 where the searcher is fixed at the center.With fast decay of the non-detection probability, detections occur early and consequently Monte Carlo runs on average end early in simulations.
We select the time step That is, the root-mean-square diffusion of the target toward the searcher in time        c r is small and the circle path covers just the area near the center, it takes long time for a target initially near the boundary to diffuse the long distance to encounter the searcher.Thus, the optimal circle path is the one that is the best compromise for taking care both the area around the center and the area near the boundary of the search region.Intuitively, one may conjecture that the optimal circle path is the one that divides the whole search region into 2 equal parts: area inside the optimal circle area outside the optimal circle.= The optimal circle radius based on the intuitive conjecture above is The results of Monte Carlo simulations in Figure 10 strongly support this conjecture.
Next we study the case of the searcher moving along a square path of half width s r with velocity 500 s v = , as illustrated in the right panel of Figure 7.
Figure 11 shows the effect of half width s r on the time evolutions of the non-detection probability.When the searcher moves along a small square path (small s r ), the non-detection probability decays moderately faster than in the case of the searcher being fixed at the center ( ) .When the square path is expanded beyond 53 s r = , the decay rate of the non-detection probability is reduced from the optimal value.When s r is large and the square path is close to the boundary of the square search region, it takes long time for a target initially near the center to diffuse the long distance to encounter the searcher.Likewise, when s r is small and the square path covers only the area near the center, it takes long time for a target initially near the boundary to diffuse the long distance to encounter the searcher.Intuitively, one may conjecture that the optimal square path is the one that divides the whole search region into two equal parts.Based on this intuitive conjecture, the optimal half width for the square path is given by  The results of Monte Carlo simulations in Figure 11 strongly support this conjecture.Before we end this section, we calculate and compare the decay rates of the non-detection probability for the three optimal cases we have considered so far.The decay rate of the non-detection probability, denoted by decay k , is calculated by fitting a straight line to data points of time vs log(non-detction probability).1) For the case of diffusing target and diffusing searcher with fixed total diffusion 100 s t D D + = , the optimal (the fastest) decay rate of the non-detection probability is achieved when the searcher is fixed at the center.The optimal decay rate is found to be

Problem 3: Searcher Moving along a Spiral
We consider the situation where the target diffuses with diffusion coefficient 100 t D = and the searcher moves along a path consisting of rotated spiral loops, which we will describe in detail below.In all simulations of the searcher sweeping a prescribed path, we use velocity 500 The selection of 500 s v = has been discussed in Problem 2. In Problem we select numerical parameters as follows: each Monte Carlo simulation is repeated 800000 N = times and the time step The increase in number of repeats from 200000 N = to 800000 N = is made possible by that the detection is faster in Problem 3, and as a result, the Monte Carlo simulations, on average, end earlier than in Problems 1 and 2.
A spiral path is a sequence of rotated spiral loops and is specified by the number of revolutions rv n in each spiral.The construction of a spiral path with 3 8 rv n = is shown in Figure 12.Each spiral loop in the spiral path is formed by a forward spiral starting at the origin and a backward spiral back to the origin.We use the Archimedean spiral.The forward spiral of rv n revolutions can be mathematically described as

rv rv rv
Let h s be the arclength of the forward spiral (half the arclength of the spiral loop).Mathematically, it fol- h rv For the forward spiral, 0 h s s ≤ ≤ .The polar coordinates ( ) , r θ of the forward spiral are expressed as functions of arclength s using the inverse function of ( ) For the backward spiral, 2 . The polar coordinates ( ) , r θ of the backward spiral are expressed as functions of arclength s as ( ) ( ) ( ) In our numerical simulations, the inverse function ( ) g − ⋅ is evaluated by solving for θ in equation ( ) The Cartesian coordinates of the spiral loop as functions of arclength s are written out based on the polar coordinates: x s r s s y s r s s Next, we scale the spiral loop formed above to fit the square search region (Figure 12(c)).We scale the spiral loop by selecting the largest coefficient b that satisfies When sweeping the spiral loop, the Cartesian coordinates of the searcher as functions of time t are given by ( .
This is the formula we use to update the searcher location in simulations.
After finishing sweeping one spiral loop, the searcher rotates the spiral loop by π 2 and starts sweeping along the rotated spiral loop (Figure 12(d)).This process is repeated until the target is detected.We point out that the optimal decay rate given above for Problem 3 is faster (larger) than those of Problems 1 and 2.

Problem 4: Searcher Moving along a Square Spiral
Now we consider the situation where the target diffuses with diffusion coefficient 100  A square spiral path is a sequence of rotated square spiral loops and is specified by the number of square layers sq n in each forward square spiral.A square spiral loop is formed by a forward square spiral starting at the origin and a backward square spiral back to the origin.
We first focus on square spirals with unit inter-layer distance.A forward square spiral of sq n square layers is  the line of angle 1 π 4 .If we do that, however, the backward square spiral will coincide with the forward square spiral in a substantial fraction of the path.Intuitively, that is not an efficient way of sweeping.We want the backward square spiral to cover the area between the layers of the forward square spiral so that together the forward and the backward square spirals have a better and more uniform coverage of the search region.We design the backward square spiral to go between the layers of the forward square spiral.Mathematically, the backward square spiral is described by As in the situation for the forward square spiral, the backward square spiral is also scaled by the inter-layer distance d given above to fit it to the search region.The backward square spiral of 2 layers ( ) After finishing sweeping the square spiral loop, the searcher rotates the whole square spiral loop by π 2 and starts sweeping along the rotated square spiral loop.This process is repeated until the target is detected.
Figure 16 plots the time evolutions of the non-detection probability when the searcher sweeps various square spiral paths.A square spiral path is specified by sq n , the number of square layers in the forward square spiral (see Figure 15).Figure 16   We point out that the optimal decay rate given above for Problem 4 is faster (larger) than those of Problems 1, 2 and 3.

Problem 5: Searcher Scanning Horizontally and Vertically
Finally we consider the situation where the target diffuses with diffusion coefficient 100 t D = , and the searcher scans horizontally and vertically back and forth.The detailed construction of scan path will be described below.The searcher moves with velocity 500 The distance from the first horizontal scan line to the last horizontal scan line is ( ) . Since the searcher is scanning a square region, we set ( ) For each forward horizontal scan, there is an associated backward horizontal scan.The backward scan travels between the horizontal scan lines of the forward scan.The forward horizontal scan is mathematically described by The associated backward horizontal scan is mathematically described by  We point out that the optimal decay rate given above for Problem 5 is faster (larger) than those of Problems 1, 2, 3 and 4.

Summary
This paper calculated the non-detection probability of a diffusing target in the presence of a stationary or moving searcher.It is found that when the searcher is fixed, the decay rate of the non-detection probability attains the maximum value when the search is fixed at the center of the square search region.When both the searcher and the target diffuse with significant diffusion coefficients, the decay rate of the non-detection probability only depends on the sum of the diffusion coefficients of the target and searcher.When the searcher moves along various deterministic trajectories, the fastest decay of the non-detection probability is obtained when the searcher scans horizontally and vertically.

Figure 1 .
Figure 1.A schematic illustration of a diffusing target in a square search region of half width A R in the presence of a searcher.The searcher may be fixed, may undergo Brownian diffusion, or may be moving with velocity s v along a prescribed path.The target is detected once it comes within distance R to the location of the searcher.

Figure 3 Figure 4
compares the non-detection probabilities obtained in 7 independent Monte Carlo simulations, each simulation consisting of 100000 N = repeats.The parameter set is the same as in Figure 2. From Figure 3, we can see that the number of repeats, 100000 N = , is large enough.Next, we explore several cases that satisfy 100 D t = 0 and the target is fixed.The initial location of the searcher is ( plots the non-detection probability for various values of s fastest decay of the non-detection probability occurs when 0 s D = (i.e., when the searcher is fixed at ( ) 0, 0 ).The decay of

Figure 2 .
Figure 2. Comparison of numerical results obtained, respectively, with time step

Figure 3 .
Figure 3.Comparison of results from 7 independent Monte Carlo simulations.Each simulation consists of 100000 N = repeats.

Figure 4 .
Figure 4. Non-detection probability for various values of s D and 100 t s D D = − .

Figure 5 .
Figure 5.Comparison of decay rates of non-detection probability for 4 sets of parameter values.

Figure 6 .
Figure 6.The effect of the searcher location on the decay rate of non-detection probability when the searcher is fixed ( ) 0 s D = .

2 t
∆ plus the distance tra- veled by the searcher in time 2 t ∆ does not exceed one twelfth of the detection radius of the searcher.We first examine the accuracy of our Monte Carlo simulations when the searcher moves along a circle of rain the left panel of Figure7.Figure8compares the non-detection probabilities obtained with two time steps:

Figure 9
compares the non-detection probabilities obtained from 7 independent Monte Carlo simulations, each consisting of 200000 N = repeats.Figure 9 demonstrates that 200000 N = is adequate for accurately capturing the decay of the non-detection probability.

Figure 10
shows the effect of the circle radius c r on the time evolutions of the non-detection probability.When the searcher moves along a small circle (small c r ), the non-detection probability decays moderately fast- er than in the case of the searcher being fixed at the center ( ) the decay rate of the non-detection probability increases.The optimal ra- dius for the fastest decay of the non-detection probability is about ( ) the circle path is ex- panded beyond 60 c r = , the decay rate of the non-detection probability is reduced slightly from the optimal value.When c r is large and the circle path is close to the boundary of the square search region, it takes long time for a target initially near the center to diffuse the long distance to encounter the searcher.Likewise, when

Figure 7 .
Figure 7.The searcher moves along a prescribed loop with velocity s v .(a) The prescribed loop is a circle of radius c r ; (b) The prescribed loop is a square of half width s r .(a) The searcher moves along a circle with velocity v s ; (b) The searcher moves along a square with velocity v s .

Figure 8 . 2 t
Figure 8.Comparison of numerical results obtained, respectively, with time step

Figure 9 .
Figure 9.Comparison of results from 7 independent Monte Carlo simulations.Each simulation contains 200000 N = repeats.

Figure 10 .
Figure 10.Results for the case of the searcher moving along a circle of radius c r .Shown here are time evolutions of non-detection probability for various values of circle radius c r .
the decay rate of the non-detection probability increases.The optimal half width for fastest decay of the non-detection probability is about ( ) optimal = 53 s r

Figure 11 .
Figure 11.Results for the case of the searcher moving along a square of half width s r .Shown here are time evolutions of non-detection probability for various values of half width s r .

2 ) 3 )
For the case of the searcher moving along a circle of radius c r with velocity 500s v =, the optimal (the fastest) decay rate of the non-detection probability is achieved when 60 c r = .The optimal decay rate is For the case of the searcher moving along a square of half width s r with velocity 500 s v = , the optimal (the fastest) decay rate of the non-detection probability is achieved when 53 s r = .The optimal decay rate is Out of these 3 cases, square loop of half width 53 s r = yields the fastest decay of the non-detection probability with decay rate 2 decay 9.88 10 k − = × .In the next section, we study the case where the searcher moves along a spiral.

Figure 12 .
Figure 12.A spiral path is a sequence of rotated spiral loops and is specified by the number of revolutions rv n in each forward spiral.A spiral path is constructed in 4 steps.(a) A spiral of rv n revolutions is used as the forward spiral in building the spiral loop; (b) The backward spiral is obtained by reflecting the forward spiral with respect to its ending angle.Together, the forward spiral and the backward spiral form the spiral loop; (c) The spiral loop is scaled to fit the square search region; (d) After finishing one spiral loop, we rotate the spiral loop by π 2 to obtain a new spiral loop.The spiral path contains these sequentially rotated spiral loops.
diagonal, pointing to a corner of the square search region (Figure12(a)). is the mirror reflection image of the forward spiral with respect to the line of ending angle (Figure12(b)).Mathematically the backward spiral is

Figure 13
demonstrates four spiral loops, respectively, of 1 .In each spiral loop, the forward spiral is shown in red and the backward spiral in blue.The correspond- ing spiral sweeping path for the searcher contains a sequence of rotations of the spiral loop.

Figure 13 .
Figure 13.Four spiral loops corresponding to 1 2 rv n = , 5 8 rv n = , 3 4 rv n = , and 1 rv n = .In each spiral loop, the forward spiral is shown in red; the backward spiral in blue.

Figure 14
Figure 14 depicts the time evolutions of the non-detection probability when the searcher sweeps various spiral paths.A spiral path is specified by rv n , the number of revolutions in the forward spiral.Figure 14 compares the results for 9 values of n rv , ranging from 1 8 rv n = to 20 rv n = .The fastest decay of the non-detection proba- bility occurs at ( ) optimal 5 8 moves along a path consisting of rotated square spiral loops, which we will describe in detail below.The searcher moves with velocity 500 s v = , the same velocity as we used in Problems 2 and 3.In Problem 4, we use the same numerical parameters as in Problem 3: each Monte Carlo simulation is repeated 800000 N = times and the time step4

Figure 14 .
Figure 14.Results for the case of searcher moving along various spiral paths.A spiral path is a sequence of ratations of a spiral loop and is specified by rv n , the number of revolutions in each forward spiral.Shown here are time evolutions of non-detection probability for various values of rv n .

Figure 15 .
Figure 15.Forward square spirals and square spiral loops.(a) The forward square spiral of 2 sq n = (2 layers); (b) The square spiral loop of 2 sq n = , formed by concatenating the for- ward and the backward square spirals; (c) The forward square spiral of 3 sq n = (3 layers); (d)

Figure 15 (
b) (blue line with filled circles).The square spiral loop is formed by combining the forward and the backward square spirals.The square spiral loop of 2 sq n = is shown in Figure 15(b).The square spiral loop starts at the origin and returns to the origin at the end.
compares the results for 9 values of sq n , ranging from 1 sq n = (the lowest possible value for sq n ) to 22 sq n = .The fastest decay of the non-detection probability occurs at ( ) The square spiral loop of 3 sq n = is shown in Figure 15(d).The corresponding square spiral path is a sequence of ratations of the square spiral loop.The decay

Figure 16 .
Figure 16.Results for the case of the searcher moving along various square spiral paths.A square spiral path is a sequence of rotations of a square spiral loop and is specified by sq n , the number of square layers in each forward square spiral.Shown here are time evolutions of the non-detection probability for various values of sq n .
velocity as we used in Problems 2, 3 and 4. In Problem 5, we use the same numerical parameters as in Problems 3 and 4: each Monte Carlo simulation is repeated scan path consists of forward horizontal scan, backward horizontal scan, forward vertical scan and backward vertical scan.A forward horizontal scan is shown in Figure 17.A forward horizontal scan is specified by 3 parameters: b the length of each horizontal scan line, d the inter scan line distance, and sc n the number of horizontal scan lines.Parameters b and d are shown in Figure 17.

Figure 17 .
Figure 17.A forward horizontal scan and associated parameters.b is the length of each horizontal scan line; d is the vertical distance between adjacent scan lines.