Wednesday, February 4, 2015

Complex Network Modeling in Maritime Search and Rescue Operation


        Complex Network is is class of complex system models which describes the dynamical set of objects. Their structure varies with time and they can be defined in probabilistic terms. Now we can find this class of complex system in various fields. Perhaps the most popular network that appears when the term 'complex network' comes is social network which may be online as well as offline. There are few others like citation networks, telecommunication networks and so on. But yes there are other systems which are similarly complex in it's structure and dynamics. One such scenario is the collective behavior of floating drifters at sea.

        Drifting floaters are one of the major subjects of Search and Rescue ( SAR ) operations. One option is a rescue of lost vessels’ crew members. Drifting and visibility are important factors for such kind of rescue operations. Another option is a search for lifeboats, washed off containers or even World War II mines. Such drift ing objects are hazardous for shipping.

         Now by floating drifters we mean a group of object that is floating in the sea with a visible distance . In context of SAR we can consider the plane wreckage floating in the sea as floating drifters after the plane is crashed into the sea.

         Let's consider there are V no of floating drifters in the sea. Each drifter can be visible within a given range. Let us consider the line of sight between two floating drifters as an edge E. Now this network evolves with time as position of each drifter is changing with the motion of waves and winds. So orientation of entire graph is getting changed with time. With the periodic motion of wave, a drifter can go out of sight and thus loosing an edge or it may comes back in sight and thus creates an edge. The set (V, E) forms a Complex Network that evolves in time. Initially the structure of Complex Network may be rather regular, but at later time the network structure is in destruction due to irregular wind, wave and current loads. 





       In the above picture we can see how the network of floating drifters have evolved over time. At time t1 it was very regular in nature and as time elapses the orientation of the network changes due to the effect of waves, winds and current. See at time t3 it becomes a graph of 2 components and in a later time they can be joined again by a link if a drifters come into the visible sight.  

         Now the sea surface can also be modeled with the superposition of finite number of harmonic waves. The disturbance of waves is determined by a two-dimensional disturbance energy spectrum S(ω,θ), where ω — the angular frequency of the wave and θ — the angle between the direction of the running wave and wind direction. 
        There are various models of the spectra of disturbance expressed in terms of the energy density function of the angular frequency and direction of propagation . An example of such a spectrum is a spectrum of Pierson-Moskowitz. The continuous spectrum of disturbance can be approximated by a finite number of harmonics. In order to provide a quick generation of harmonic heights and velocities, the two-dimensional Fast Fourier Transform (FFT) can be used. With the help of all these mathematical tools the planer displacement and velocities of floating drifters and water particles can be obtained. These attributes help us to understand the dynamics of the floating drifters. 

Application: In Search and Rescue Plans

     Considering all the factors including direction of wind, wave, gravity the probable trajectory of the floating drifters can be approximated which helps in rescue operation. Because for example when a plane crashes in mid air and all its wreckage falls into the sea from that high altitude then the wreckage will be spread across miles over the sea. So the helicopter needs to follow the direction of the main wave initially and after that it needs to predict to some extent about the probable location of other parts of the plane considering all the natural parameters like wind, sea wave, current etc.

  Again from the above picture say for example due to strong waves some containers have been washed away from the vessel. And after a time the crew has noticed this and sends a helicopter for the rescue of the containers. Now the helicopter starts to fly in the direction of the main wave with a fixed velocity until a cargo is spotted. The helicopter then continues the search for another closest visible cargo or it can again go in the direction of the main motion. Now it should be remembered that high velocity and large visibility range does not guarantee containers rescue because greedy algorithm can run into wrong decision that is critical for Travelling Sailsman Problem.


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