Need for Optimal . . . Case of Multi-Zone . . . Case of Moving Boundary Towards Optimal How to Arrange . . . Sensor Placement . . . Sensor Placement in Where to Place . . . Where to Place . . . Conclusion: . . . Multi-Zone Measurements Conclusion: Resulting . . . Octavio Lerma, Craig Tweedie, Title Page and Vladik Kreinovich ◭◭ ◮◮ Cyber-ShARE Center ◭ ◮ University of Texas at El Paso El Paso, TX 79968 Page 1 of 22 Go Back Full Screen Close Quit
Need for Optimal . . . 1. Outline Case of Multi-Zone . . . Case of Moving Boundary • In multi-zone areas, boundaries change with time. How to Arrange . . . • It is desirable to place sensors in such a way that the Sensor Placement . . . boundary is covered at all times. Where to Place . . . • In this talk, we describe the optimal sensor placement Where to Place . . . with this property. Conclusion: . . . Conclusion: Resulting . . . • In this optimal placement, sensors are placed along a Title Page see-saw trajectory between ◭◭ ◮◮ – the current location of the boundary and ◭ ◮ – its farthest future location. Page 2 of 22 Go Back Full Screen Close Quit
Need for Optimal . . . 2. Need for Measurements Case of Multi-Zone . . . Case of Moving Boundary • In many areas, we know the partial differential equa- How to Arrange . . . tions that can be used for the prediction. Sensor Placement . . . • Example: weather prediction. Where to Place . . . • To make accurate predictions, we need to have a very Where to Place . . . accurate picture of the initial conditions. Conclusion: . . . Conclusion: Resulting . . . • This picture must describes current values of Title Page – temperature, ◭◭ ◮◮ – atmospheric pressure, ◭ ◮ – wind, Page 3 of 22 – and other characteristics Go Back at different spatial locations. Full Screen • To measure the values of these characteristics, we need Close to place sensors at different locations. Quit
Need for Optimal . . . 3. Need for Sensor Placement Case of Multi-Zone . . . Case of Moving Boundary • In some areas – e.g., in and around big cities – there is How to Arrange . . . usually a large number of sensors. Sensor Placement . . . • Many of the sensors operated by volunteers who place Where to Place . . . these sensors in their homes. Where to Place . . . • However, in other areas (e.g., in the Arctic), the exist- Conclusion: . . . ing sensor coverage is too sparse. Conclusion: Resulting . . . Title Page • So, more sensors are needed. ◭◭ ◮◮ • The placement of new sensors ◭ ◮ – not only helps in achieving short-term goals such Page 4 of 22 as weather predictions, Go Back – it also helps in analyzing long-term effects such as climate and environmental changes. Full Screen Close Quit
Need for Optimal . . . 4. Need for Optimal Sensor Placements Case of Multi-Zone . . . Case of Moving Boundary • Placement and maintenance of sensors in remote areas How to Arrange . . . is often costly. Sensor Placement . . . • So, it is desirable to come up with the optimal ways to Where to Place . . . place sensors – so that we can achieve Where to Place . . . – the desired accuracy and Conclusion: . . . Conclusion: Resulting . . . – coverage at the smallest possible cost. Title Page • The problem of optimal sensor placement is simpler for ◭◭ ◮◮ homogeneous (single-zone) regions. ◭ ◮ • In these region, the measured quantity smoothly changes Page 5 of 22 from one location to another. Go Back • The i -th sensor measures the value v i = v ( x i ) at its location x i . Full Screen • For locations x close to x i , we have v ( x ) ≈ v ( x i ) = v i . Close Quit
Need for Optimal . . . 5. Optimal Sensor Location: Case of Single-Zone (Ho- Case of Multi-Zone . . . mogeneous) Regions Case of Moving Boundary How to Arrange . . . • Suppose that we have placed sensors at locations x 1 , . . . , x n . Sensor Placement . . . • For every spatial location x , we approximate v ( x ) by Where to Place . . . the result v ( x i ) measured by the closest sensor i . Where to Place . . . • The quality of a sensor network is measured by accu- Conclusion: . . . racy with which it determines v ( x ) at all x . Conclusion: Resulting . . . Title Page • From this viewpoint, the quality of a sensor network is ◭◭ ◮◮ determined by the “worst” spatial location x 0 . ◭ ◮ • x 0 is a spatial location which is the farthest away from all the sensors. Page 6 of 22 • For location x 0 , the approximation accuracy is the worst. Go Back Full Screen • Thus, for homogeneous regions, the optimal sensor place- ment is uniform. Close Quit
Need for Optimal . . . 6. Case of Multi-Zone Measurements Case of Multi-Zone . . . Case of Moving Boundary • In practice, regions are often not homogeneous. How to Arrange . . . • Regions consist of several distinct zones with a sharp Sensor Placement . . . boundary between the zones. Where to Place . . . • The simplest example is a shoreline – the boundary Where to Place . . . between the land and the ocean. Conclusion: . . . • In mountain regions, there is also a sharp lines between Conclusion: Resulting . . . Title Page a glacier and the grassy zone around it, etc. ◭◭ ◮◮ • For such multi-zone areas, ◭ ◮ – we need not only to find the values of the desired characteristics at different zones, Page 7 of 22 – we also need to get a good understanding of the Go Back exact location of the boundary between the zones. Full Screen • In practice, the boundaries change. Close • It is very important to trace these changes. Quit
Need for Optimal . . . 7. How to Optimally Place Sensors Near the Inter- Case of Multi-Zone . . . Zone Boundary Case of Moving Boundary How to Arrange . . . • We assume that the boundary is reasonable smooth. Sensor Placement . . . • So, for some reasonably large (and known) value ℓ , Where to Place . . . – if we place sensors at distance ℓ from each other, Where to Place . . . – then we will get a pretty good picture of the whole Conclusion: . . . boundary. Conclusion: Resulting . . . Title Page ✛ ℓ ◭◭ ◮◮ ✲ ℓ ℓ ✏✏ ✶ ✐ P P ✏✏✏✏✏ PPPPP ✏ PP ✮ ✏ q r r ◭ ◮ r r Page 8 of 22 • Each sensor is located at a distance ℓ from the previous Go Back one. Full Screen • We need to cover the boundary of total length L . Close • So, we need L/ℓ sensors. Quit
Need for Optimal . . . 8. Case of Moving Boundary Case of Multi-Zone . . . Case of Moving Boundary • We must cover not only the current boundary, but also How to Arrange . . . its future locations. Sensor Placement . . . • So, we need to place several lines of sensors. Where to Place . . . • Each sensor line requires L/ℓ sensors. Where to Place . . . Conclusion: . . . • Let N denote the total number of sensors that our Conclusion: Resulting . . . budget can afford. Title Page • Thus, we can place k = N/ ( L/ℓ ) sensor lines. ◭◭ ◮◮ • Let V 0 be the speed with which the boundary moves. ◭ ◮ • Let T be the planned lifetime of the sensor network. Page 9 of 22 • Thus, we need to place k sensor lines at distances from Go Back 0 to D = V 0 · T . Full Screen • It is reasonable to select these distances to be equally Close spaced, at distance 0, d = D/k , 2 · d , 3 · d , . . . Quit
Need for Optimal . . . 9. How to Arrange Different Sensor Lines Relative to Case of Multi-Zone . . . Each Other? Case of Moving Boundary How to Arrange . . . • On each sensor line, the sensors are equally spaced, Sensor Placement . . . with a distance ℓ between two neighboring sensors. Where to Place . . . • Once we determine a place for one of the sensors on Where to Place . . . this line, the location of others is determined. Conclusion: . . . • Namely, we place sensors on this line at distances ℓ , Conclusion: Resulting . . . 2 · ℓ , . . . , from this original sensor. Title Page ◭◭ ◮◮ • Let us start with such an equally spaced arrangement of sensors on the original boundary. ◭ ◮ • Let us pick two neighboring sensors at distance ℓ from Page 10 of 22 each other. Go Back • For each sensor line, we can then select the segment Full Screen parallel to the segment between these two sensors. Close Quit
Need for Optimal . . . 10. Sensor Placement (cont-d) Case of Multi-Zone . . . Case of Moving Boundary • On a segment of length ℓ , each sensor line has exactly How to Arrange . . . one sensor. Sensor Placement . . . • Once the locations of all these sensors is fixed, the lo- Where to Place . . . cation of all other sensors is uniquely determined. Where to Place . . . • On each sensor line, we place sensors on this line at Conclusion: . . . distances ℓ , 2 · ℓ , . . . , from the sensor from this segment. Conclusion: Resulting . . . Title Page • Thus, to fully determine the sensor configuration, we ◭◭ ◮◮ must decide how to place sensors within this segment. ◭ ◮ r r r ✻ Page 11 of 22 r r r Go Back D r r r Full Screen r r r ✻ d Close ❄ ❄ ✛ ✲ r r r ℓ Quit
Recommend
More recommend
