Ciat Access

Accessibility Modelling in ArcView 3An extension for computing travel time and

market catchment information

By

Andrew Farrow and Andy Nelson

2001

Accessibility Modelling in ArcView 3 CIAT

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Contents

Acknowledgements

Installation

Introduction

Description

Example

References

ArcView Technical Notes

Acknowledgments

• The data used in the examples were taken from the CIATHonduras database. More details can be found here.http://www.ciat.cgiar.org/scale/data

• The Technical Notes section has been adapted from thedocumentation provided with ArcView 3.2 .

• Several parts of the Accessibility Analyst have been based onscripts found at the ESRI ‘Arcscripts’ websitehttp://www.esri.com/arcscripts

Script AuthorCost Distance Grid Tools ESRIGraphics.Graphics2Shapes [email protected]

Oregon Department of ForestryMake Extension Tools ESRIClipping Grids Tom Van Niel

• CIAT donors generously provided the funding for thedevelopment of this software. In particular we would like tomention

The Ecoregional Trust Fund for Methodological DevelopmentThe World BankUNEP

• The following people at CIAT have contributed to the ideas thatlead to this software and manual.

http://www.ciat.cgiar.org/scale/data

http://www.esri.com/arcscripts

mailto:[email protected]


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Andy Farrow, Andy Nelson, Manuel Winograd, Jeremy Eade,Gregoire Leclerc, Ron Knapp and William Bell.

Installation under Windows

This extension requiresArcview 3 and the Spatial Analyst extension

1 Unzip the file CIAT_access.zip.

2 Place ciat_access.avx in the extension directory.

You may need administrator privileges to do this.

3 Place the ciat_access directory anywhere you like.It contains the manual and the example datasets.

4 Start ArcView and the extension will be available in theExtensions dialogue as Access Analyst (Ver. 2.0)

Installation under UNIX

This extension requires Arcview 3 and the Spatial Analyst extension

1 Uncompress CIAT_access.Z.

$> uncompress CIAT_access.Z

2 Place ciat_access.avx in the extension directory.Assuming that your system has the environment variable$AVHOME.

$> cp access.avx $AVHOME/ext

You may need to be superuser to do this.

Alternatively, if your system has the environment variable$USER_EXT, you can place ciat_access.avx in thatdirectory.

3 Place the ciat_access directory anywhere you like.It contains the manual and the example datasets.

4 Start ArcView and the extension will be available in theExtensions dialogue as Access Analyst (Ver. 2.0)


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Introduction

Accessibility

Transportation is a critical function for an economy as it affects themovement of people, goods and services, and development. In mostcountries, a complex transportation network permeates thecountryside with paths for walking, paths for horses and donkeys,single lane seasonal roads, one and two lane all-weather gravelroads, local surfaced roads, regional and national surfaced roads,and motorways or freeways with dividers separating thecarriageways. Each of these has typical speed of travel associatedwith it, which can be adjusted to take into account slope, the type ofvehicle traversing it, rainfall over a 2, 3 or 7 day period, or anadjustment for congestion. In the end it is possible to work out howlong it takes to get from anywhere on the map to the nearest markettown or hospital or school using a particular vehicle class. The timetaken to reach a desired location can be generally referred to as its‘accessibility’.

Accessibility can be defined as the ability for interaction or contactwith sites of economic or social opportunity, (Deichmann, 1997),however there are a multitude of ways in which this intuitive concepthas been expressed in the literature. Goodall (1987) definesaccessibility as the ease with which a location may be reached fromother locations, and Geertman (1995) states that the concept ofaccessibility can be used in rural development policy as an indicatoror rural deprivation and as a variable on location analysis

Over the last ten years there has been a growing awareness thatrural accessibility concerns more than just roads, (Barwell, 1996).Reports and surveys carried out in developing regions paint a pictureof rural isolation and unproductive use of limited resources, in which

the farmer largely inhabits a walking world. In his introduction toTransport and the Village, Cleaver states that:

It is clear that the extremely poor state of off the road transportsystem in Africa acts as a powerful brake on agriculturalproductivity and growth. Improved accessibility will reduce theeconomic costs of moving goods from local markets and easethe barriers to social facilities. This will contribute to economicgrowth and enhance social well-being.

But before national governments can provide environmentsconducive to the development of local infrastructure, there is a needfor a model that can incorporate the local environmental factors thatdefine rural inaccessibility. Also any model should be flexibleenough not only to quantify time and cost of travel but alsoinfrastructure improvement (or degradation) scenarios to be createdand evaluated.

Clearly there are different levels of infrastructure availability,transport availability, and many different environmental effects, andeconomic considerations. For example it would be impossible toapply the same model of rural accessibility to Sub Saharan WestAfrica and Central American Hillsides, but any model should becapable of incorporating all of these factors in a way that is not onlysensible and geographically sensitive but also educational andexplanatory.

BackgroundSpatial accessibility is determined by geographic location in relationto target locations, and by the transportation facilities that areavailable to reach those destinations. Accessibility is also influencedby social factors such as knowledge and information and byeconomic factors since the use of transport and communicationfacilities is usually associated with some monetary cost. Although itis difficult to incorporate these economic factors explicitly in spatialmodels, the selective weighting of one market can imply the socialand economic dimensions over another.


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In public planning, an important objective is the equal provision ofservices for all people and in all parts of a country. For example, thepopulation of a country or region should have similar access to publicservices, independent from their location of residence. Poorinfrastructure and consequently limited accessibility implies feweropportunities for improving economic status, health or socialstanding. Ravallion (1996), for example, speaks of a “spatial povertytrap” that may prevent the poor from breaking out of local levelconstraints.

Operational quantitative measures of accessibility are thereforeuseful in policy analysis where the focus is on service provision, forexample, in the health sector. Other areas in which accessibility isimportant are economic applications, and, in fact, many of theoperational measures of accessibility have first been developed forlabour market research and retail analysis.

Geographical information systems (GIS) lend themselves naturally tothe computation of accessibility indicators. GIS can representnetworks, villages or facilities and provide functions to computedistances and to define relationships among spatial objects.Consequently, certain accessibility measures can be computed usingpackages such as IDRISI, GRASS and ArcView.

However, there is a distinct lack of any step-by-step, or easy to usemodels to develop transport maps, accessibility surfaces and theeconomic catchment areas of towns and markets. This is a strangeomission from the planner's toolbox, when accessibility is consideredsuch a fundamental indicator of economic potential and anundeniable pressure on ecological systems. One possibleexplanation is that accessibility is so poorly defined; many peopletalk about accessibility and there are many differing opinions as towhat it actually is. The following section describes a simple butflexible GIS method and tool for deriving verifiable accessibilitymodels.

Description

To begin with, it is necessary to clarify some of the terminology thatwill be used in this model. Accessibility is calculated on a frictionsurface. A friction surface consists of a regular two-dimensional gridwhere each cell in the grid represents either a transport route suchas roads, railway lines, tracks, or navigable rivers, or relativelyinaccessible land and water bodies.

Different types of transport infrastructure have differentcharacteristics. A surfaced road, for example, allows faster travelspeed than a dirt road. In practice, it is therefore not enough tomeasure the distance of a road connection between two points.Instead, a measure of travel cost is preferable. This cost can bemeasured in monetary terms or as travel time. As will be discussedlater in this section the cost to travel across different types of landcan be easily estimated.

Targets (places of interest) such as villages, hospitals, or nationalparks are usually located on the transport network and can thereforebe represented in another grid as cells that have a certaincharacteristic. This characteristic could be the village’s population,the hospital’s number of doctors or beds, or the number of teachersat a school. If we are simply interested in is the presence orabsence of a place of interest, then all cells can have the samevalue.

All that is needed to create an accessibility map are:

• A coverage of markets or places of interest

• A grid where each cells value represents the cost oftraversing that particular cell.


These two datasets are input to a COST-DISTANCE model, which isdiscussed in more detail in the Technical Notes section.

So far the problem seems trivial, the cost-distance functionin concept and the data requirements are minimal. In fact two of the reasons for selecting this type of analysis. Howeare several points to note.

1. You need geographic data that is in an EquProjection (for example Lambert Azimuthal), to areal qualities.

2. You are unlikely to have a friction surface ready to h

3. The friction surface is totally dependent on the upurpose, and will possibly change for differing sci.e. foot based/vehicle based, wet season/dry seaso

4. The pre-processing required to create the eqprojected grids.

With that in mind, an interface was developed at CIAT to petype of spatial analysis quickly and intuitively.

ArcView GIS was chosen as the working environmenactivity. Previous work in CIAT had used ArcInfo and its amacro language AML to produce accessibility surfacesversion 3, ArcView had become mature enough development of the interface using its Avenue scripting lang

The ArcView extension addresses all of these issues, with a simpledefinition or concept of accessibility.

What is the cost of getting from any locationto the nearest place of interest?

This is a fairly general question, where cost can be:

• Real cost (cost of transport)

• Perceived cost (cost of transport with other factors)

• Time (simply time taken to get there)

• Perceived time (travel time plus factors introduced by thenature of the transport)

• Any other concept of cost that the user is able to express

A place of interest can be:

This is a key point. The ArcView GIS and ArcInfo tnotes, state that any data that are to be used as inthe cost-distance algorithm should be in an equprojection. For very small regions or areas nEquator, this may not be so critical.

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• A market where goods can be bought or sold (large towns)

• A service industry (so called enterprise zones)

• A hospital or school

• A national park or any other area under threat by humandevelopment

The location can be anywhere in the area of study.


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The following section is a step by step introduction to the ArcViewextension.

Example

Load the Access Analyst Extension

In the extension dialogue, select Access Analyst from the list. Thiswill load both the Access Analyst and Spatial Analyst extensions.

Open a new view

The Accessibility menu will be added to the View menu bar. Thereare two options in the menu, English or Español. Select yourpreferred language to start the analysis. The Accessibility Dialoguewill appear, asking for a directory in which to place all temporarygrids and coverages. You will not be able to move on to the nextstage until you have entered a valid directory. Once a directory hasbeen entered, click Next.


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Add friction themes to the view

A new View called Friction Surface Themes will be created. At leasttwo datasets are needed for creating the friction theme; a transportor road cover, and a cover that defines the limit of the analysis.These two must be loaded before any optional friction componentscan be included. Clicking the buttons will open the Add Theme pop-up window and you will be prompted to add the themes.

The text alongside each button will confirm that the theme has beenloaded. You can always change the themes by clicking therespective theme button to delete the current choice and selecting anew one. Once these two themes have been added, you can eitherclick Next to proceed or you can add optional themes by clicking thecheck box. In this example, we will add more friction themes.

There are five options available,

• Rivers• Slope• Land Cover• Urban Areas• Barriers

Depending on the analysis, you may want to include one or more ofthese themes as a contributing factor to the friction surface. In thisexample we will add Rivers, Slope and Land Cover.

When a theme is loaded, a new x button appears alongside, which ifclicked will delete the respective theme allowing you to replace it withanother choice.


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Once all the optional fiction themes have been added, click the AddOptional Themes check box to hide the optional themes, and thenclick Next. Use the Back button if you want to go back to a previousstage at any time in the analysis

Add a source or target theme to the viewThe final necessary theme is the source or target point theme, i.e. adataset that contains all the points to calculate accessibility to. Asbefore, click the Add Target Theme button to load a source/targetdataset. Once loaded, the Make a Selection check box becomesvisible. If you do not wish to alter the points that you have loadedyou can just click Next to proceed, but if you wish to make aselection, click the check box.

There are three options for making sub selections on the pointdataset.

• Select from Table• Select from Query• Select from the Map

Here we will make a sub selection based on Table of the dataset.Clicking, Select From Table brings up the table, record can beselected which will be highlighted (usually in yellow) in the view. Inthis case we select two points out of the sixteen.

Once a selection has been made, click the check box once again tohide the options and click Next.


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Clip all themes to a common spatial extentBy now there are at least three themes in the view and perhaps asmany as eight. In the event that these themes do not share acommon spatial extent, it is necessary to clip them.

A pop-up window will explain that a red box will appear in the viewthat can be interactively resized should you wish to clip the themes.Click OK and the red box will appear. You can resize the box untilyou have focused on your area of interest. When ready, click ClipThemes. If some of your themes are complex or big, this can takesome time, this is a good time for a coffee break.

If you do not wish to clip the themes, leave the box in its initialposition and simply click Clip Themes.

All the clipped themes will be transferred to a new View calledClipped Themes. Now click Next to proceed to the next stage.

Convert all themes to gridsThe cost-distance algorithm will only work with grid datasets, so thisstage converts all the datasets to grids. A pop-up window will ask forthe analysis properties to be set, the most important property beingthe cell-size. In this case we can set the cell-size to be equal to theslope grid (100m), which is also an easy number to use for thecalculations that have to be made in the latter stages. When the cell-size has been set, click OK in the pop-up to start the grid conversionprocess. All the grids will appear in a new View called GriddedThemes.


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For each grid, you will be asked to select the item from the table touse for the grid values. Select whichever value is most appropriate,and select No, when asked to join attributes to the grid.

When all the themes have been converted, click Next.

Reclass the grids to reflect their friction valueIn this stage all the grids must be reclassed so that the value of eachcell represents the cost or time required to traverse the cell. Forexample with a grid consisting of roads, we might estimate that a carcan travel at 60km/hr. so all road cells in a 100m grid would be giventhe value 6

60 km/ hr = 60,000 m/hr = 1,000 m/min so 100m takes 6 seconds.

Or for land cover, average walking speed is estimated at 4km/hr, butis reduced to 3km/hr for forest, giving values of 90 and 120respectively.

4 km/ hr = 4,000 m/hr = 67 m/min so 100m takes 90 seconds.3 km/ hr = 3,000 m/hr = 50 m/min so 100m takes 120 seconds.


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In this case these values have to be calculated for each

• road type• river type• land cover type (background value)• boundary (this value is not important)

In this case we have small cell sizes so the cost will be calculated inseconds, but it could also be minutes or hours. So the generalconversion factor for turning km/hr into seconds is.

1Time = Cell size x

1000Speed(km/hr) x

3600

Slope is a special case where we need to calculate a slope factor,i.e. how much does slope affect the speed of travel over the othersurfaces. For example slopes between 0 and 5 degrees may nothave any influence so the factor would be 1, but slopes between 5and 10 degrees would slow travel by half so the factor would be 2,etc. There is no accounting for slope direction, it is assumed thattravelling both up-slope and down-slope incurs a reduction in travelspeed.

Click on the Reclass button to start and use the pop-up windows toreclass each grid. All the grids will appear in a new View calledReclassed Themes. If you wish to change the classification of any ofthe grids, click Reclass again to repeat the process. When all thethemes have been reclassed, click Next.

Combine the friction grids into oneThe cost-distance algorithm requires only two inputs, a friction themeand a source theme. In this stage, all the friction components(roads, rivers, boundary, etc.) are merged to create one grid. Clickthe Combine button to do this. When the Friction grid appears, clickNext to continue

The grids are merged using the following logic, cell-by-cell.


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for each in_cell{

if in_cell = barrierout_cell = barrier x slope

else if in_cell = roadout_cell = road x slope

else if in_cell = riverout_cell = river

else if in_cell = urban areaout_cell = urban area x slope

else if in_cell = land coverout_cell = land cover x slope

}

Run the Cost-Distance functionClicking the cost-distance will start the algorithm. Three grids will becreated.

• allocation grid - Indicating the catchment area of each target• direction grid - Indicating the direction of travel through each

cell• time to market grid - Indicating the cost of travel from each cell

to the nearest target (target defined by the allocation grid)

Here we are showing the time to market grid that was calculated inseconds but has been reclassed into minutes for clarity. We havealso loaded the two targets into the view. Click Next to continue.


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Convert outputs to ShapefilesThe final stage allows the user to convert the three outputs toshapefiles. If the study area is large or the time to market surface iscomplex, this may take a long time. Optionally you might want toreclass the grids (seconds to minutes or minutes to hours forexample) and then convert them to shapefiles.

A composite map of time to markets, the transport network and allpopulated places in the vicinity of Yorito and Sulaco in Honduras.


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BibliographyArchondo-Callao R. and Faiz A. (1994), Estimating vehicle-operating costs. WorldBank technical paper No. 234. pp 82.

Barwell I. (1996), Transport and the village. Findings from Africa-level travel andtransport surveys and related studies. World Bank Discussion Paper No. 344. pp 66.

Chesher A. and Harrison R. (1987), Vehicle Operating Costs. Evidence fromdeveloping countries. The Highway design and maintenance standards series. WorldBank Publication. pp 374.

Deichmann U. (1997a), Accessibility and Spatial Equity in the Analysis of ServiceProvision, Workshop on Geographical Targeting for Poverty Reduction and RuralDevelopment, The World Bank, Washington, D.C., November 11, 1997.

Deichmann U. (1997b), Accessibility Indicators in GIS, United Nations StatisticsDivision, Department for Economic and Policy Analysis, New York.

Deichmann U. and Bigman D. (2000), Spatial indicators of access and fairness forthe location of public facilities. In Bigman, David and Hippolyte Fofack (Editors)Geographic targeting for poverty alleviation, methodology and application. WorldBank Regional and Sectoral Studies, Washington DC.

Eade J. (1997), The accessibility surface. Electronic Mail [email protected], to [email protected] (Accessed Feb 21, 1997).CIAT, Cali, Colombia.

Geertman S.C.M. and Reitsema van Eck J.R. (1995), GIS and models ofaccessibility potential: An application in planning. International Journal ofGeographical Information Systems 9, 67.

Dixon-Fyle K. (1998), Accessibility planning and Local Development. InternationalLabour Organisation. Technical report.

Kammeier H.D. (1999), New tools for spatial analysis and planning as components ofan incremental planning-support system. In: Environment and Planning B, vol 26, pp.365-380.

Leinbach T. (1995), Transportation and Third World Development: Review, Issues,and Prescription. Transportation Research. Vol.29A, No.5, pp. 337-344.

Ravallion M. (1997), Good and Bad Growth: The Human Development ReportsWorld Development, Vol. 25, No. 5, pp. 631-638.

Ritsema van Eck J.R. and de Jong T. (1999), Accessibility analysis and spatialcompetition effects in the context of GIS-supported service location planning. In:Computers, Environment and Urban Systems, volume 23, pp. 75-89.

Teeffelen van P. Barkhof M. and de Jong T. (2000), Applying a GIS for integratedaccessibility planning in Laos using a PC based GIS -FLOWMAP- as a tool to planfacilities. Fifth seminar on GIS and developing countries GISDECO, International RiceResearch Institute, Laguna, Philippines.

Winograd, M. Farrow, A. and Eade J. (1997), Environmental and sustainabilityindicators outlook for Latin America and the Caribbean. CD-ROM with manual.UNEP-CIAT, Cali, Colombia. pp 27

Winograd M. Fernandez N. and Farrow, A. (1998), Herramientas para la toma dedecisiones en AmericaLatina y el Caribe: indicadores ambientales y sistemas deinformacion geograficos/ Tools for Making decisions in Latin America and theCaribbean: environmental indicators and Geographical Information Systems. CIATand the United Nations Programa de Naciones Unidas para el Medio Ambiente(PNUMA).

Wolff de T. Kruska E Ouma E. Thornton P. and Thorpe W. (2000), Improving GISderived measures of farm market access: An application to mil markets in the EastAfrican highlands. Fifth seminar on GIS and developing countries GISDECO,International Rice Research Institute, Laguna, Philippines.




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ArcView Technical Notes

aGrid.CostDistance(costGrid,direction,allocation,maxDistance)

A Spatial Analyst function that calculates for each cell the least-accumulative cost-distance over a cost surface, costGrid, to cellswith values other than No Data in aGrid, optionally producingdirection and allocation grid data sets (direction and allocation).

direction can be a FileName or Nil, and defines the FileName for theoptional cost direction grid data set. allocation can be a FileName orNil, and defines the FileName for the optional cost allocation griddata set. If either direction or allocation is set to Nil the optionaloutput grid data set for the argument that is set to Nil is notcalculated.

maxDistance can be a Number or Nil, and defines the threshold thatthe accumulative cost values cannot exceed. If a cell exceeds thevalue it is given the value of No Data. If maxDistance is set to Nil,then the value of infinity is used.

This request will trigger the evaluation of aGrid, if aGrid has not yetbeen evaluated. For more information on the evaluation of Gridobjects, see the Discussion topic for the Grid class.

This request is only available with the Spatial Analyst.

Cost Distance Modelling DiscussionThese functions are similar to Euclidean distance functions, butinstead of calculating the actual distance from one point to another,they determine the shortest cost distance (or accumulated travelcost) from each cell to the nearest cell in the set of source cells. Asecond exception is that cost distance functions apply distance not ingeographic units but in cost units.

All cost distance functions require a source Grid and a cost Grid. Asource Grid can contain single or multiple zones, which may or maynot be connected. All cells that have a value (including 0) areprocessed as source cells. All non-source cells need to be assignedNo Data on the source Grid.

A cost Grid assigns impedance in some uniform-unit measurementsystem that depicts the cost involved in moving through anyparticular cell. The value of each cell in the cost Grid is assumed torepresent the cost-per-unit distance of passing through the cell,where a unit distance corresponds to the cell width. These costs maybe travel time, dollars, preference and so forth.


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The Cost CalculationsThe aGrid.CostDistance request creates an output Grid in whicheach cell is assigned the accumulative cost to the closest sourcecell. The algorithm utilizes the node/link cell representation. In thenode/link representation, each centre of a cell is considered a nodeand each node is connected by links to its adjacent nodes.

Every link has impedance associated with it. The impedance isderived from the costs associated with the cells at each end of thelink (from the cost surface) and from the direction of movement. Ifmoving from a cell to one of its four directly connected neighbours,the cost to move across the links to the neighbouring node is 1 timesthe cost of cell 1 plus the cost of cell 2 divided by 2.

a1 = cost1 + cost2 / 2

where cost1 is the cost of cell 1, cost2 is the cost of cell 2 and a1 isthe length of the link from cell 1 to cell 2.

The accumulative cost is determined by the following formula.

accum_cost = a1 + (cost2 + cost3) / 2

where cost2 is the cost of cell 2, cost3 is the cost of cell 3 andaccum_cost is the accumulative cost to move into cell 3 and cell 1.

If the movement is diagonal, the cost to travel over the link is1.414216 (or the square root of 2), times the cost of cell 1 plus thecost of cell 2 divided by 2.

a1 = 1.414216(cost1 + cost2) / 2

But when determining the accumulative cost for diagonal movementthe following formula must be used.

accum_cost = a1 + 1.414216(cost2 + cost3) / 2


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The AlgorithmCreating an accumulative cost-distance Grid using graph theory canbe viewed as an attempt to identify the lowest cost cell and adding itto an output list. It is an iterative process that begins with the sourcecells. The goal of each cell is to be assigned quickly to the outputcost-distance Grid.

In the first iteration, the source cells are identified and assigned tozero since there is no accumulative cost to return to themselves.Next, all the source cell’s neighbours are activated and a cost isassigned to the links between the source cells nodes and theneighbourhood cell’s nodes using the above accumulative costformulas. Each of these neighbourhood cells can now reach asource, consequently, they can be chosen or assigned to the outputaccumulative cost Grid. To be assigned to the output Grid, a cellmust have the next least-cost path to a source.

The accumulative cost values are arranged in a list from the lowestaccumulative cost to the highest.

The lowest cost cell is chosen from the active cost list and the valuefor that cell location is assigned to the output cost-distance Grid. Thelist of active cells is now expanded to include the neighbours of thechosen cell, because those cells now have a way to reach a sourceOnly those cells that can possibly reach a source can be active in thelist. The cost to move into these cells is calculated using theaccumulative cost formulas.


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Again, the active cell on the list with the lowest cost is chosen, theneighbourhood is expanded, the new costs are calculated and thesenew cost cells are added to the active list

Source cells do not have to be connected. All disconnected sourcescontribute equally to the active list. Only the cell with the lowestaccumulative cost is chosen and expanded, regardless of the sourceto which it will be allocated.

Furthermore, cells on the active list are updated if a new, cheaperroute is created by the addition of new cell locations to the outputGrid.

This updating can occur with the advent of new paths for cells on theactive list as more cells are allocated to the output Grid. When thecell with the lowest value on the active accumulative cost list isallocated to the output Grid, all the accumulative costs arecalculated. These costs are also calculated for the neighbouring cellsof the newly assigned output cell, even if the neighbouring cells areon the active list through another cell. If the new accumulative costfor the locations on the active list is greater than the one that thecells currently has, the value is ignored. If the accumulative cost isless, then the old accumulative cost for the location is replaced onthe active list with the new value. That cell, which has discovered acheaper and more desirable path to a source, then moves up on theactive chosen list. In the example below, the cell location at row 3,column 1 (highlighted by the box) had an accumulative cost of 11.0when it was put on the active list to reach the source at the top of theGrid. But because the lower source expanded to this location, thecell had access to a cheaper accumulative cost path to reach asource. The value for the location was updated on the active list andallocated to the output earlier, because of this lower accumulativecost.

If there are multiple zones or disconnected sets of source cells onthe input source Grid, the growing process continues and allocatesthe cheapest cost cell from the active list, regardless of which source


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it is from. When the growth fronts meet, the least cost path back tothe source proceeds until all eligible cells have received a cost value.

t is conceivable that when the growing patterns meet, cells from onegrowth pattern will discover that they can reach a source cell inanother set or growth pattern more cheaply; if so, they will bereassigned to the new source. This behaviour was witnessed by thecell at row 3, column 1 earlier, but is also exemplified below by thecell located at row 3, column 6.

When all cells have been chosen from the active list, the result is theaccumulative-cost or weighted-distance Grid. The procedure usedensures that the lowest accumulative cost is guaranteed for eachcell.


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Cost Back LinkThe cost-distance Grid identifies the accumulative cost for each cellto return to the closest cell in the set of source cells. It does not showwhich source cell to return to or how to get there. The cost back linkreturns a Grid with a value range from 0 to 8 that can be used toreconstruct the route to the source. Each value (0 through 8)identifies which neighbouring cell to move into to get back to thesource. 0 is a source, 1 indicates move to the immediate right, 2 tothe lower right diagonal, 3 to the cell immediately south, etc..

If the cell is assigned 5 as part of the least cost path to a source, thepath should move to the left neighbouring cell. If that cell has 7, thepath should move due north.

Cost AllocationCost allocation produces a Grid very similar to the Euclideanallocation function; like the Euclidean function, it returns a Grididentifying which cells will be allocated to which source, but unlikethe Euclidean function, on the basis of the lowest accumulative costto reach a source.

Cost PathOnce the accumulative cost and back-link Grids are created, least-cost path routes can be derived from any designated destination cellor zone(s). The cost-path request retraces the destination cellsthrough the back-link Grid to a source. If there are multiple cells orzones as input destinations, the least cost path can be calculatedfrom each cell (resulting in multiple paths, one path for each cell), oreach zone (one path from each zone).


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When two or more cost paths from different zones converge on theway to a source and continue the remaining distance together, thejoint path to the cell is assigned 2, since the owner of the segmentcannot be determined. The value assigned to each path is assignedby the order in which the zone cells are encountered in the scanningprocess.

The set of source cells consists of all cells in the source Grid thathave valid values. Cells that have No Data values are not included inthe source set. The value 0 is considered a legitimate source. Cell

locations with No Data in the cost Grid act as barriers in the costsurface functions. Any cell location that is assigned No Data on theinput cost surface will receive No Data on all three output Grids ofthe aGrid.CostDistance request. If a mask has been set in theanalysis environment, any masked cells will be treated as No Datavalues.

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