These pages will explain how to do network analysis in raster GIS using MFworks as an example software.

MFworks has evolved from MAPFactory, originally designed by C. Dana Tomlin, the father of map algebra. This is unmistakably seen in MFworks command procedures and scripting language. MFworks is thus the ideal companion for teaching and learning Tomlin’s principles of map algebra.

MFWorks was featured in a very positive review in the June 2000 edition of GeoEurope as the only raster GIS truly capable of network analysis.

Conducting network analysis in MFworks comprises iterative steps that lead to a functioning network. These steps will convert map layers with square cells into linear elements that are linked together as lines, with directional flows assigned to each cell, and map layers containing cost variables.

This example, based on my MSc in GIS, will show how network analysis is performed in MFworks, both with a fixed travel cost for the entire time of travel and with a dynamic travel cost, where the cost of travel changes during the time of travel.

The key to producing a successful network model is in understanding the relationship between the characteristics of physical network systems and the representation of those characteristics by the elements of the network model. The efficacy and validity of the network depends on how precisely the network can be modeled to match the real world network it represents. A network can be explicitly modeled in vector GIS, but can only be approximated by a raster GIS like MFworks.

Network representation in vector GIS: Nodes (numbers) and arcs (letters)

A network in vector GIS takes the form of edges (or arcs) connecting pairs of nodes (or vertices). Nodes can be junctions and edges can be segments of a road. For a network to function as a real-world model, an edge will have to be associated with a direction and with a measure of impedance, determining the resistance or travel cost along the network.

Even if it does not appear so explicitly, a grid made up of cells in a raster GIS is in fact a graph representing a network, with 8 possible directions from each node. However, the grid cells only approximate the exact shapes of the lines in the network. Also, the direction is not as explicitly given as in the vector model. Thirdly, the line and node attributes must be stored as a separate layer for each attribute. As a result, a network using a raster model normally consists of a vast number of layers. Since the grid has a given resolution, the cells will only approximate the exact length of the network.

Tracing a path from cell to cell in raster GIS
generates a zigzag path instead of a straight line.

Using Tomlin’s Incremental Length, Incremental Linkage, and Directional Identifiers, which identify underlying linear features, MFworks makes it possible to model a road network in raster GIS in much the same manner as in vector GIS. The first step is to extract linear features from neighboring raster cells; the second step is to assign directions.

This operation infers the lineal characteristic of raster cells, by equating consecutive locations with a set of straight lines between them (Figure 1-3). Based on its relations with neighboring cells that have the same attribute value, each cell is given a linkage value indicating how it is linked to other cells.

Incremental Linkage, cell value infers the linear structure it represents.
© Thinkspace

By assigning a value to each cell equivalent to the linear feature it represents it is possible to create a network similar to a road network. The smaller the cell resolution, the better the real-world road network will be approximated by this procedure.

Building a road network using Incremental Linkage from
figure 1-3 with cell values representing linear features

The second step to creating a road network in raster GIS is to impose constraints on the flow that can take place from cell to cell.

The second step to creating a road network in raster GIS is to impose constraints on the flow that can take place from cell to cell. The value assigned to the centre cell in a 3×3 window indicates the directions the flow can take in and or out of this cell. Figure 1-5 shows how a cell value of 10 is inferred from flow in direction 8 and 2.

Tomlin’s directional identifiers: Cell values indicate possible flow direction in or out of cell

The directional identifiers that are to be assigned to any given cell in a road network can be directly inferred from the Incremental Linkage values, i.e. Incremental Linkage value 28 yields directional constraints value 10, and so on. The transition from Incremental Linkage to Directions is done through a straightforward reassigning of the cell values in the Incremental Linkage map layer to corresponding values in the Directions map layer. More specific constraints, like one-way directions or dead-end roads, which are not directly inferable from the mentioned linkage values, will have to be assigned manually.

Inferring flow directions from Incremental Linkage values in figure above

Usually, to generate a cost-of passage surface, several variables will be collapsed into one layer. These variables might be road class, average speed, traffic density, and congestion during specific time of day or other factors that contribute to the overall cost variable. The cost-of-passage surface can be defined by a variety of measurement units: time, fuel consumption, money or other possible cost units, for which the least cost passage is to be determined.

Using average speed and time as a means of inferring cost-of-passage is among the most common approach in network analysis, since it is easy to use and calculate. However, “least cost” does not always need to be “least time”; it may just as well be least fuel, least length, or any least cost variable that can be implemented in a cost-of-passage surface.

Cost surface, with values indicating impedance to travel (cost) across the cells.

To determine the actual length of a path through a number of cells the Incremental Length operation is used. Incremental Length works similar to Incremental Linkage to the extent that Incremental Linkage is used implicitly to determine the linkage, from which the length is inferred. Incremental Length then applies the factor by which the cell resolution has to be multiplied to yield the length of the linear features in any cell.

Incremental Length cell values indicate factor for calculating length of linear features. © Thinkspace

It should be noted that Incremental Length calculates the factor for inferring the total length of all linear features in any cell. When uncritically applied to deriving the time of traversing a cell, this function may not yield exact results, since the time of passing straight through a cell may differ from the calculated value, which takes into account all linear features in that cell, even those that are not traversed. The smaller the cell resolution and the higher the average speed, the more negligible this error becomes.

Assessing discrepancy between actual path length
and length inferred from Incremental Linkage

The figure and table above demonstrates an example of this “error”. The bold line is the path sought; the shaded cells indicate where the Incremental Length factor deviates from the actual length. The table shows how cell resolution and average speed influence the magnitude of the error.

To determine the length of a particular road stretch, this stretch first has to be separated, then made subject to the Incremental Length operator to find the correct length. Whether this last procedure is absolutely necessary will depend on the cell resolution and whether deriving the actual path length is strictly required as a result of the task in question.

A special case, illustrating the discrepancy between inferred and actual path length, as described previously, appears at crossroads and junctions and deserves particular attention, especially at non-90-degrees junctions, as shown in the following example:

Sought path through a junction

The actual path sought is a straight line through the junction, indicated by the shaded cells. The bold lines indicate the actual linear features. Here, the actual path length is cell resolution x 5.6 (1.4 x 4).

Inferred path, as result of Incremental Length/ Incremental Linkage

In order to justify the inferable directions, the Incremental Length operator will link to many cells together, as shown by the shaded cells and bold line. The inferred path is equal to cell resolution x 8.2 (1.7 x 2 + 1.4 x 2 + 1 x 2), thus overestimating the actual path over this section by 2.6 x cell resolution. Unless there is substantial difference in cost-of-passage between the two roads that cross each other, which then prohibits the off-path cells from being included in the added proximity surfaces, this error will occur in most cases with such junctions.

Correctly modelled junction, linkage indicated by bold lines, direction indicated by arrows

A solution to avoid this from the beginning, before finding the least-cost path, would be to model the linkage correctly, and allow the direction to flow differently. The fact that it is possible to completely detach the direction of the flow in a network from the underlying linear feature is indeed an interesting observation concerning the modeling of networks in raster GIS.

The above can also be demonstrated by an example from the study area. Note the linked square in the left image, as opposed to the straight line in the right image:

(a) Least-cost path using generically inferred directions

(b) Location of a and c in study area

(c) Least-cost path using manually inferred directions

One option for implementing a dynamic cost surface is to use a continuously updated network, where the latest available data at the starting time of travel is used, and where the cost of travel does not change during the duration of the estimated route. Another option is to establish a network with varying travel cost per pre-defined time interval. In this case the travel cost is dependent on the starting time of travel, and the travel cost changes when the estimated route passes from one time interval to another.

In practice, the first approach involves building an application that passes on values to the cost surface used in a raster-based GIS for calculating the fastest path in a network. The second approach involves working with multiple cost surfaces for estimating the fastest path, which is the one that will be used here.

This approach uses an average cost-of-passage surface for each time interval. If one-hour time intervals were used, one would need 24 cost surfaces to cover a 24-hour period.

Applied within the context of time intervals, the first proximity surface is cut off when the time interval border is traversed. Then, from all endpoints, new proximity surfaces and shortest paths are calculated, until the destination is reached. Computationally this can turn into a considerable task, depending on the number of time intervals traversed and the number of paths generated in each time interval: If time interval A yields 4 possible paths from a starting point, then for time interval B, the four paths must be continued until the destination or a new time interval is reached. When the destination has been reached the values of all paths must be added and compared. In practice, one must calculate proximity surfaces from a starting point to cut-off points where a new time interval begins, and from cut-off points to starting point, adding the surfaces yields the accumulated cost-of-passage (time) value for each path. Then, from each cut-off point the procedure has to be repeated until destination or a new time interval is reached. Finally the accumulated cost-of-passage value for each successively linked path must be calculated and compared to find the path with the lowest accumulated value.
In detail, the steps would be as follows:

Input: A network, with 3 possible ways from starting point to destination point.

Step1: Spread from starting point, cut-off when accumulated cell values indicate that a time interval borderline is being passed. Then, spread backwards from the cut-off points to starting point. Adding the proximity surfaces yields the cost-of-passage for this time interval for the various paths.

Step 2: For all cut-off points, spread until next time interval or (in this case) until the destination has been reached. Spread backwards from destination and add proximity surfaces to yield the cost-of-passage values for all possible paths. Retain the paths from the cut-off points to destination that have the lowest cost-of-passage.

Discard other paths

Step 3: Add the cost-of passage for the partial paths to yield the final cost-of-passage value for the remaining paths.
Retain the path with the lowest value.

Accuracy

Using raster GIS for network analysis leads into a simplification of a complex network structure. The path is prone to be distorted, firstly, due to the mistaken length introduced by Incremental Length, and secondly, due the zigzagged path, a consequence of the innate grid structure. Nonetheless, a fine-tuned use of Incremental Linkage and a minimal cell resolution can have a smoothing effect on the exact delineation of the path. On the other hand, minimising cell resolution increases computation.

In raster GIS the precision of the model is determined by the cell resolution, the finer the resolution (the smaller the cell size), the better the precision. For this research a cell resolution of 20m was deemed appropriate for the task in question, as it would allow encompassing normal road width and adjacent areas within one cell width. For a dense inner city road network a cell resolution of 10m would be preferable, allowing even narrow blocks to be incorporated into the model.

The infamous multiple path problem

The concept of adding proximity surfaces will some times lead to peculiar artefacts. A special case appears in a regular lattice if the cost surface is isotropic: Here the multiple path problem is clearly visible, since all possible paths per se have the same accumulated cost towards the destination. Here, human mind, as opposed to computer mind, would intuitively seek out a solution that implements a heuristic search, always following the path that yields the shortest Euclidian distance to the destination.

Multiple path problem in regular lattice with homogenous cost surface. Start at top left, destination at bottom right.

More often occurs the so-called multiple path problem, particularly if the road network and the cost surface are oversimplified. There are no distinctive low values, so that in order to gain a coherent route from start to destination so many low valued cells have to be selected that no clear path can be discerned.

Typical multiple path problem. Start at top left, destination at bottom right.

Map layers needed: A network, a cost surface, origin and destination point,

network.mfm

costoftravel.mfm

startstop.mfm

(image missing)

The network is a replica of the Brown’s Pond study area used by Tomlin. The cost surface are fictitious values, 1-4, indicating impedance. Origin and destination were derived by assigning the values 999 and 998 respectively to points on the network map (startstop.mfm) and then extracting each point using Recode.

linkage = IncrementalLinkage network;

The Linkage operation assigns values indicating how cells are linked together.