A Fast Method for Calculating Emergency Response Times Using Travel Resistance Surfaces

An algorithm is described that quickly calculates minimum travel times between locations for initial response forest fire suppression units. The algorithm was developed for integration into wildfire planning simulation models to quickly identify fire suppression units with the fastest response to a fire during a simulation. While generally quite fast, network-based path minimization algorithms are not suitable for the wildfire problem since fires are often located away from existing network features, such as roads and trails. On the other hand, traditional raster-based least-cost path algorithms can take minutes, if not hours, to calculate each resource’s travel time, too slow to be effective within a simulation model. The algorithm presented in this paper enhances the raster-based technology, providing a multi-resolution approach. Our test results show the new algorithm dramatically increases the speed of calculations with only limited loss of accuracy Introduction One vexing problem confronting the design of realistic simulation models for initial attack wildfire planning is accurately estimating ground suppression resource travel times between locations. From the standpoint of initial attack simulation model building, there is more than just a need for accurate estimates of travel times; there is also a need within the simulation model to quickly make these calculations thereby maintaining reasonable simulation run times. In this paper, we describe a computationally efficient shortest-path algorithm for computing travel times within an initial attack wildfire simulation model. This algorithm, COSTPATHSQ (Q for quick), was developed to meet the needs of the Wildfire Initial Response Assessment System (WIRAS), a comprehensive wildfire initial attack planning tool (Wiitala and Wilson, this proceedings). 1 An abbreviated version of this paper was presented at the second international symposium on fire economics, planning, and policy: a global view, 19–22 April 2004, Córdoba, Spain. 2 GIS Developer, Digital Visions Enterprise Unit, Forest Service, U.S. Department of Agriculture, 333 SW 1st Ave, Portland, OR, 97204. Operations Research Analyst, Forestry Sciences Laboratory, Forest Service, U.S. Department of Agriculture, 620 SW Main Street, Suite 400, Portland, OR 97205. Resource Information Specialist, Forestry Sciences Laboratory, Pacific Southwest Research Station, U.S. Department of Agriculture, 620 SW Main Street, Suite 400, Portland, OR 97205. GIS Analyst, Digital Visions Enterprise Unit, Forest Service, U.S. Department of Agriculture, 1230 NE Third St., Suite A-262, Bend, OR 97701. USDA Forest Service Gen. Tech. Rep. PSW-GTR-xxx. xxxx. 2 Proceedings of the Second International Symposium on Fire Economics, Planning, and Policy: A Global View GENERAL TECHNICAL REPORT PSW-GTR-208 Session Poster— A Fast Method for Calculating Emergency Response Times —Hatfield, Wiitala, Wilson, and Levy WIRAS simulates the effectiveness of a ground and air resource suppression organization in fighting forest wildfires. During a simulation, WIRAS needs to evaluate the opportunities for dispatching up to 100 suppression resources to numerous fire locations. This need for evaluation can happen many times during a single simulation leading to tens of thousands of travel time estimates. To be effective, the WIRAS simulation model requires computations to occur within a fraction of a second while providing a limited loss of accuracy on estimates of travel time. To accomplish this objective, we first explored the possibility of computing travel times over a network using the Dijkstra shortest-path algorithm (Dijkstra 1959). While numerous enhancements have been made over the years to the Dijkstra algorithm (Zahn 1997), for our particular problem, tests using the Dijkstra algorithm as implemented within ArcInfo (ESRI 2001) proved much too slow. Therefore, we decided to develop COSTPATHSQ, a new variation on the implementation of the Dijkstra algorithm that could exploit the particular nature of our travel time problem and meet the computational requirements for the WIRAS simulation model. Methods COSTPATHSQ uses the Dijkstra algorithm to estimate the least-cost path, where cost refers to travel time in this instance. In our model, travel is performed on a raster cost surface or grid rather than a vector network because wildfires do not necessarily fall on network features, such as roads and trails. A network could be constructed across the whole landscape, but the file size would be excessive and we believe the processing time would be prohibitively slow. A raster cost surface can be viewed as a regular triangular network with a network node at the center of every cell in the raster and network links radiating from each node to each of the eight adjacent nodes, horizontally, vertically, and diagonally. The main enhancement to the Dijkstra algorithm described here is the construction and employment of a multi-resolution cost surface, or cost pyramid. The cost pyramid allows COSTPATHSQ to initially and quickly move across a coarse resolution surface. The algorithm employs finer resolution surfaces to more accurately locate the end points (fig. 1). Depending on computational tolerance and accuracy needs, the user of COSTPATHSQ determines the number successive levels of resolution desired. Figure 1—How least-cost paths are constructed using a cost pyramid. USDA Forest Service Gen. Tech. Rep. PSW-GTR-xxx. xxxx. Proceedings of the Second International Symposium on Fire Economics, Planning, and Policy: A Global View 593 Session Poster— A Fast Method for Calculating Emergency Response Times —Hatfield, Wiitala, Wilson, and Levy Cost Pyramid The COSTPATHSQ algorithm determines for a cost surface the least-cost path between points using the Dijkstra algorithm in a manner similar to its implementation in ArcInfo (ESRI 2001). As mentioned above, COSTPATHSQ enhances the Dijkstra algorithm by employing the use of a multi-resolution cost surface, or cost pyramid. The cost pyramid is constructed by successively aggregating blocks of four cells in an upward progression through the surfaces (fig. 1). The first or bottom level of the pyramid is constructed of the original cells in the cost surface. These cells are grouped into fours to form new cells on the second level of the pyramid. Within each second-level cell, the first-level cell that is deemed to be the best pathway is selected as the center or hub of the second-level cell. The cost to travel between adjacent second-level cells is calculated between the cells’ hubs using the Dijkstra algorithm to travel on the first level of the pyramid. These eight travel costs are stored in each of the cells in the second level of the pyramid. Subsequently the second-level cells are grouped into fours to form cells on the third level of the pyramid (the highest level in fig. 1). Within each third-level cell, the second-level cell that is deemed to be the best pathway is selected as the hub, and the cost between third-level cells is calculated using the Dijkstra algorithm to travel on the second level of the pyramid. This process is repeated until only two rows and/or columns of cells remain. The result is increasingly generalized cost surfaces that retain much of the accuracy of the source data. This system of cost surfaces allows COSTPATHSQ to estimate the leastcost path between points in a fraction of a second with a limited loss of accuracy. Pathways The determination of which cells make better pathways is done prior to constructing the pyramid and is based upon the underlying transportation network. It is accomplished by selecting the one cell from the entire raster that bears the lowest cost and traveling the least-cost path to this origin cell from every other cell in the cost surface. If more than one cell bears the lowest cost, then one of these cells is selected at random to be the origin cell. The result of this preprocessing is a pathway grid. The value of each cell in the pathway grid is the number of times that that cell is crossed when traveling from each cell in the grid to the origin cell. Cells that receive higher values are deemed to be better pathways. Thus when four first-level cells are grouped to form a second-level cell, the first-level cell with the highest value in the pathway grid is selected as the hub of the second-level cell. This maximum value is assigned to the second-level cell and is used during the determination of hubs on the third-level of the pyramid, and so on.