# Value of Information Based Data Retrieval in UWSNs
## 3 Value of Information in Underwater Sensor Networks
### Temporality of VoI; Infotentials  
Infotential is a term that we have coined for information potential. By potential we imply the potential of impact an event has on an asset, and therefore, the report of such an event can be deemed to have a certain information potential. Infotentials are functions that encode the variability of VoI.  
Vol of a report is subject to various factors such as the valuation of the asset being reported; the impact of the event being reported on the asset; timeliness of the report; reliability, precision, and accuracy of the report; and the fact that whether this report is a repetition or not. Asset valuation dictates Vol but as there is a temporal dynamism to this valuation, therefore, Vol should correlate with this variability across time. We quote two factors that result in asset valuation decay; the first is that assets naturally depreciate with time; the other is that events of catastrophic nature can result in a sharper decline in the valuation of an asset. As both these fiscal attributes are decaying in nature, therefore, Vol should be a monotonically decreasing function.  
In the context of this paper, we define a function that encodes the temporal variability of Vol. We consider this variability to be monotonically decreasing in time and assume that at any point in time the valuation of the asset does not increase due to any other factors. These functions can be designed in a variety of different ways so as to fit the needs of an application scenario. We model depreciation with a decaying exponential so as to maintain generality. Otherwise, functions such as descending staircases or any other complex combination of exponentials can also be used. Ideally, the function should be constructed based on practical statistical data.  
In this study, we employ exponential functions of the form  
\[v(t)=Ae^{-B(t-\tau_{o})} \tag{1}\]  
Examples of the infotential \(v(t)\) are shown in Figure 2a. The parameters that control the function are \(A\) and \(B\) and \(\tau_{o}\). The parameter \(\tau_{o}\) is the time at which the event is reported. Parameters \(A\) and \(B\) scale the Vol across the domain and range of the function. Parameter \(A\) represents the valuation of information at \(t=\tau_{o}\) while parameter \(B\) represents the decay in information valuation \(A\) for \(t>\tau_{o}\), i.e., after the onset of the event being reported at \(\tau_{o}\). The two exponential functions in Figure 2a could be understood as representing two different classes of information in an application scenario. The more the valuation of an asset is, the higher the number \(A\) would be. Likewise, the higher the \(B\) value is, greater is the rate of damage to an asset.  
Let there be two events \(E_{H}\) and \(E_{L}\) with significance and damage rate \(\{S_{H},D_{H}\}\) and \(\{S_{L},D_{L}\}\) respectively. The subscript \(H\) corresponds to a high-priority event while the subscript \(L\) indicates a low-priority event. We denote the corresponding Vol functions as,  
\[v_{H}(t) =A_{H}e^{-B_{H}(t-\tau_{o}h)} \tag{2}\] \[v_{L}(t) =A_{L}e^{-B_{L}(t-\tau_{o}l)} \tag{3}\]  
The relationship between significance and valuation, and between damage rate and decay, is given as,  
\[S_{H}>S_{L} \implies A_{H}>A_{L} \tag{4}\] \[D_{H}>D_{L} \implies B_{H}>B_{L} \tag{5}\]  
These inequalities are general guidelines. The actual values of parameters \(A_{x}\) and \(B_{x}\) need to be configured with the help of a system expert or should be based on statistical data.