Optimal Deployment Problems of Radar Network

Abstract: The deployment problems of the radar sets are important in the air defense of the military. The target detection joint probability of several radar sets is studied. The optimum deployment models of circle, line and sector have been built up. The area of space, which is determined by the detection joint probability is more than 0.9, is got by the use of Monte Carlo method. The optimum deployments problems of the circle deployment, line deployment and sector deployment can be solved by entire enumerate method, golden section method and coordinate alternation method. On condition that the number of radar is less in the twice-ines deployment model it can be solved by combine entire enumerate method with coordinate alternation method. Contrarily, it can be solved by combining genetic algorithm with coordinate alternation method.


INTRODUCTION
The modern air attack environment is getting more and more complex.Attackers may fly at very low to very high altitude and they may be bombers, interceptors, or air to ground missiles etc.It is impossible that a single radar set has a satisfactory capability against all possible threats.So several radar sets are deployed to detect the attackers (Zou and  Chakrabarty, 2004).The optimal deploy problem of two sets of radar is solved by the use of Monte Carlo method (Gao, 1999).Various radar and jammer parameters for effective luring away of the missile are studied (Vijaya et al., 2011).In this study, the target detection joint probability of several radar sets is studied and the optimum deployment models of circle, line and sector have been built up.

DETECTION PROBABILITY OF SINGLE RADAR SET
Detection probability of single radar set is AFSC TR-65-2 (1965) and Nelson (1954).The detection probability is computed using (1), as the parameter of radar and target is given.As the parameter of radar and target is known, ap T /2z 2 is a fixed number.If we assume k = ap T /2z 2 , the detection probability of target relates only to target distance.It can be calculated simply as: (2) In general, the value of k can be got by measurement.Suppose that detection probability of target at range of 32 km on the surface is 0.684 by testing.Applying (2) result in: k = -In 0.316×32 4

DETECTION JOINT PROBABILITY OF SEVERAL RADAR SETS
If all radar sets are independent to each other, the detection joint probability can be obtained as follows (Ravindran and Phillips, 1987; Zou and Chakrabarty,  2003): where, p i = Detection probability of radar i where, (x i , y i ) is coordinate of radar i and (x, y, z) is coordinate of target.
The detection joint probability at a definite height is discussed mainly.The detection joint probability at altitude z = h 0 can be expressed as:

MONTE CARLO METHOD
The detection joint probability must be high (for example p L ≥ 0.09) enough to detect target.The space determined by p L ≥ p 0 can be expressed as: Now key problem is how to deploy radar sets as to get area of space determined by p L ≥ p 0 maximum.
It is difficult to compute area by multiple integral method.We use Monte Carlo method to resolve this problem.The general algorithm follows: Step 1: Set N = 10000, m = 0 and j = 0.
Step 3: replace m s by m + 1. Otherwise return to step 2.
Step 4: Let S ≈ 4m/ N X max Y max (Where S = area of space determined by p L ≥ p 0 maximum) and stop.

THE OPTIMUM DEPLOYMENT MODEL OF THE CIRCLE
The optimum deployment models are many, such as the circle, line and sector deployment.Suppose that several radar sets are deployed on the circle (Fig. 1).The radius is r.The coordinate of radar i is: The detection joint probability of typical target at altitude z = 10 km can be obtained using (5).The space determined by p L ≥ p 0 maximum can be also obtained using (6).The area can be estimated by Monte Carlo method, as r 0 from 0 to 120 km and step is l km.We can obtain the maximum area.Table 1 show the solution.The relation between area of detection with r when n = 9 be shown in Fig. 2. The shape of the maximum area can be shown in Fig. 3.The number of S max / n will increase according as n increase.In other words, the average effectiveness of radar will increase according as the number of radar set increase.But as radar sets are deployed on the circle, the detection joint probability is lower than 0.9 at its centre (Fig. 2).

THE OPTIMUM DEPLOYMENT MODEL OF THE LINE
Suppose that several radar sets are deployed on the line (Fig. 4).The interval is d, This line is axis X and the center of radars is zero.The coordinate of radar i is The area can be estimated also by Monte Carlo method, as d from 0 to 120 km and step is l km.But the runtime is long.We can use golden section method to solve this problem.The golden section method is described as follows: Step 1: Set α = 0, b = 100 and ε > 0 is given.
Table 2 shows the solution.The maximal area of detection when n = 3, z = 10 km and d = 56.6 km is shown in Fig. 5.

The Optimum Deployment Model of the Sector:
Suppose that several radar sets are deployed on the sector (Fig. 6).The angle is α.The coordinate of radar i is: Table 3 shows the solution by golden section method.The maximal area of detection when n = 3, z = 10 km and r = 58.3km is shown in Fig. 7.

THE TWICE-LINES DEPLOYMENT MODEL
In order to defend important spot, we can deploy the radar sets in twice-lines manner.Suppose that several radar sets are deployed on the sector in multilines (Fig. 8).
Suppose the inner radius is r 1 and the outer radius is r 2 (r 2 > r 1 ) .The inner numbers of radar sets is n 1 and the outer numbers of radar sets is n 2 (n 2 ≥ n 1 ) n 2 .The coordinate of radar i is: The space determined by p L ≥ p 0 maximum can be also obtained using (6).Dimension of determined area is S(n 1 , n 2 , r 1 , r 2 ).The optimal model is 1 . .,there are four situation (n 1 , n 2 ) = {(1,7), (2,6), (3,5), (4,4) }.We simulate respectively and find out the optimal deployment.If the n is big, the genetic algorithm is used to solve it.
• If r 1 , r 2 , n 1 , and n 2 is unknown, the model ( 11) is a hybrid nonlinear programming model.If the n is big, we can use the entire enumerate method with coordinate alternation method.When n are known, we can use the combining genetic algorithm with coordinate alternation method.

CONCLUSION
The optimum deployment models of circle, line and sector have been built up.The other deployment can be deal with similarly.The model of detection probability of radar is simple in this study and the precise model will be study in the future.The multilines deployment model can be solved similarly.
probability a = Available reflection area of target z = Standard deviation of noise amplitude p T = Power of transmitter r = Target distance Fig. 1: Circle deployment

•
Fig. 8: Twice-lines deployment• If r 1 , r 2 , are known and the n is not big, we can use entire enumerate method.Given a group (n 1 , n 2 ), we can got the dimension by Monte Carlo method.For example, as 8 = n