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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.

Search problems arise commonly in many diverse areas [

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. [

In [

Consider a square region with half width

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

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

and consider five problems below.

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

In our numerical discretization,

In Monte Carlo simulations, we advance the target and the searcher in time according to

where

where function

It is straightforward to verify that

The target is labeled as “detected” at

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

For each set of parameter values, we repeat the Monte Carlo run

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

We first examine the accuracy of our Monte Carlo simulations in the case of

Next, we explore several cases that satisfy _{t} = 0 and the target is fixed. The initial location of the searcher is

the non-detection probability is slowed down when the searcher diffuses with

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

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.

Next we study the case of a fixed searcher

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

Next, we study the case where the searcher moves with a constant velocity

We consider the situation where the target diffuses with diffusion coefficient

Let

The distance traveled by the searcher with velocity

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

We pick

In all the simulations below, we use

In Problem 2, for each set of parameter values, we repeat the Monte Carlo run

We select the time step

That is, the root-mean-square diffusion of the target toward the searcher in time

We first examine the accuracy of our Monte Carlo simulations when the searcher moves along a circle of radius

The optimal circle radius based on the intuitive conjecture above is

The results of Monte Carlo simulations in

Next we study the case of the searcher moving along a square path of half width

The results of Monte Carlo simulations in

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

1) For the case of diffusing target and diffusing searcher with fixed total diffusion

2) For the case of the searcher moving along a circle of radius

3) For the case of the searcher moving along a square of half width

Out of these 3 cases, square loop of half width

In the next section, we study the case where the searcher moves along a spiral.

We consider the situation where the target diffuses with diffusion coefficient

The increase in number of repeats from

A spiral path is a sequence of rotated spiral loops and is specified by the number of revolutions

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

We select the starting angle

Recall that in our problem, the searcher moves along a path with a constant velocity. To use the spiral loop as the searcher’s sweeping path in simulations, we need to express

where function

Let

For the forward spiral,

For the backward spiral,

In our numerical simulations, the inverse function

The Cartesian coordinates of the spiral loop as functions of arclength s are written out based on the polar coordinates:

Next, we scale the spiral loop formed above to fit the square search region (

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

_{rv}, ranging from

We point out that the optimal decay rate given above for Problem 3 is faster (larger) than those of Problems 1 and 2.

Now we consider the situation where the target diffuses with diffusion coefficient

In Problem 4, we use the same numerical parameters as in Problem 3: each Monte Carlo simulation is repeated

A square spiral path is a sequence of rotated square spiral loops and is specified by the number of square layers

We first focus on square spirals with unit inter-layer distance. A forward square spiral of

described as follows. We cycle through 4 directions: positive x, positive y, negative x, negative y. If we start at

With this simple notation, the forward square spiral of

In this unit forard square spiral, the inter-layer distance is 1. Next we scale the unit forward square spiral to fit it to the search region. Let d be the inter-layer distance after the scaling. We select the inter-layer distance d as

where

We could select the backward square spiral as the mirror image of the forward square spiral with respect to

the line of angle

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

rate of the non-detection probability at

We point out that the optimal decay rate given above for Problem 4 is faster (larger) than those of Problems 1, 2 and 3.

Finally we consider the situation where the target diffuses with diffusion coefficient

In Problem 5, we use the same numerical parameters as in Problems 3 and 4: each Monte Carlo simulation is repeated

The scan path consists of forward horizontal scan, backward horizontal scan, forward vertical scan and backward vertical scan. A forward horizontal scan is shown in

A forward horizontal scan is specified by 3 parameters: b the length of each horizontal scan line, d the inter scan line distance, and

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

The forward horizontal scan and the associated backward horizontal scan for

The vertical scans are exactly the same as the horizontal scans except that the roles of x and y are swapped. The forward vertical scan is

The associated backward vertical scan is

The forward vertical scan and the associated backward vertical scan for

To scan the square search region of half width

Thus, for a square of given half width

The searcher sequentially cycles through forward horizontal scan, backward horizontal scan, forward vertical scan and backward vertical scan. This process is repeated until the target is detected.

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.

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.

Hong Zhou would like to thank Professor James Eagle, Professor Sivaguru Sritharan and Professor Jim Scrofani for stimulating discussions. The views expressed in this document are those of the authors and do not reflect the official policy or position of the Department of Defense or the US Government.