Dasymetric Mapping

Some geographical distributions are best mapped as ‘volumes’ that represent surfacescharacterized by plateaus of relative uniformity separated from one another by relatively steepslopes or escarpments where there is a marked change in statistical value.

There are two techniques for defining this stepped surface, the choroplethic and thedasymetric. The choroplethic technique requires only grouping of similar values, and detail isconstrained by the boundaries of enumeration units which rarely have to do with the variable beingmapped. In choroplethic mapping, emphasis in the graphic statement is placed upon comparingrelative magnitudes across the surface of the map. In contrast, the dasymetric method highlightsareas of homogeneity and areas of sudden change and is produced by refining the values estimatedby the choroplethic technique.

Your task is to produce a dasymetric map of cropland in south central Ohio counties, usingfour variables to refine the enumerated data. Prepare this map and all related maps usingArcView GIS. It might be a good idea to keep a journal log while you are working through thisexercise, annotated with hardcopy maps if you prefer, personal notes describing in your own wordswhat the commands you performed in ArcView do, for later reference.

Data for the initial distribution is given on the base map on page 4. Initial files for theassignment are in the Handouts folder in the usual Csiss folder on \\ubar\labs\. Copy the folderlabeled dasy_ohio from the Handouts directory to your personal directory (for example to E:).Open the file dasy.apr in your directory with ArcView. When opening the project file you mightbe asked where certain missing files are. If so, select the files in your directory that have the samename as the missing ones, and click each time OK to continue. Once all the data is reassignedyou should see an open view, labeled Cropland in Ohio 1959 with the five themes describedbelow. Go to FILE: SET WORKING DIRECTORY... type in:your_local_directory:/your_personal _directory/dasy_ohio/ to set your working directory toyour home directory and the lab folder you copied over. ArcView will produce a bunch of filesand you want to make sure that are all stored in the lab directory. Make sure that any folder orfile associated with this lab in your directory does not have any spacing in the label, norUppercase.

Please Note: You might not be able to finish this lab in one sitting. So, don’t worry if youdon’t. You can find the result maps at the end of this handout just in case.

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For much of the exercise, the following classes will be used:

0 – 9% coverage of cropland10 – 29%30 – 49%50 – 69%70 – 89%90 – 100%

On any maps you print out, include the following information:• an appropriate title• a legend• a scale• your name• data source and year

Use partial spectral progressions to indicate the classes of your map and for all worksheet mapsexcept for economic activity which should be mapped using a full spectral progression. We havebriefly touched on this concept in lectures. Part spectral progressions are generated as lineartransects across or through the color wheel; full spectral progressions are generated as arcs of full orpartial circumference. Think about the layer you are mapping (e.g. coverage by woodland,coverage by urban areas, terrain configuration, and coverage by cropland) and apply a meaningfulprogression.

The following materials are provided to help you complete this exercise. The files arescanned or digitized from 1:100,000 scale copies of 1:24,000 maps. The stated resolution (.60km per cell) is meaningful, and should guide your decisions about how much detail to include orsimplify.

This handoutA discussion of the dasymetric technique and the methods to employ, including step-by-stepprocedures for preparing a dasymetric map.

Maps

• choropleth % crop A base map displaying the percentage of total area classed as croplandfor each county. The single figure for each county includes several classes of cropland: croplandharvested, cropland used only for pasture, cropland not harvested and not pastured, areas in grassesand legumes for soil improvement, and idle cropland (fallow) and crop failure. For each county,these values have been summed and the total divided by total county acreage to give thepercentage. The remaining area in each county consists of urban and rural non-agricultural landuses,woodland (pastured and non pastured), and non urban built-up land (e.g. farmsteads). Data arederived from the 1959 Census of Agriculture.The county names are included in the theme’s attribute table. In addition, there is a theme withcounty labels called county names.

• towns A map of urban and rural non-agricultural land use areas. The areas outlined areoccupied by residential, commercial, industrial, transportation, mining and similar land uses.Areas of many sizes have been shown, from the completely built-up city and metropolitan areas tocrossroads settlements, hamlets, and galaxies of strip mines which dot the Ohio countryside. The

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areas are shown as close to scale as possible, but crude reproduction techniques do involve someexaggeration. These data are derived from 1:24,000 topographic sheets. Maps from the 1960Census were used to outline the urban areas.

• woodland A map showing the amount of woodland cover throughout the area in six classes.It was derived from the same topographic sheets and reflects a series of estimates based on USGS7.5 minute quadrangles.

• terrain A ‘surface configuration’ map shows in four classes the nature of the terrain. This mapwas generalized using the work of Guy-Harold Smith, “The Relative Relief of Ohio”,Geographical Review vol.25, p. 272-284, 1935.

• economy A map of ‘economic regions’ outlines areas by agricultural economic type. This isbased on work by Alfred J. Wright, “Types of Farming Areas”, Economic Geography of OhioColumbus Division of the Geological Survey of Ohio, Bulletin #50, 1953, Figure 13.

It is important to realize that the concern in this exercise is with the dasymetric procedure and notwith cropland in Ohio. Since this exercise marks the first time that you are presented with thisparticular mapping technique (map overlay) the methods are set down in considerable detail.After gaining understanding of the nature of the technique from the following section, the simplestway to proceed is simply to follow the steps in the order of presentation. Once accomplishedhere, map overlay techniques as used in conventional geographical information systems or forspatial modeling should be more readily understood.

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The Dasymetric techniqueThe dasymetric technique provides one solution to the problem of mapping data gathered on thebasis of enumeration areas whose boundaries bear no direct relation to the variable being mapped.It is based on certain assumptions.

Assumption 1 the variable being mapped occurs non-uniformly over the statistical unit area (inthis case, counties).

Assumption 2 even though wide overall variations exist for the distribution, it basically consistsof areas of relative uniformity separated by sudden changes in value.

Assumption 3 other variables may be collected in association with the variable in question whoserelation to the variable may be determined and expressed as a set of rules. Thesevariables will enable the cartographer to adjust and refine the given data to formhomogeneous regions whose boundaries are independent of the enumeration unitboundaries.

The variable in question for this exercise is cropland (density). The four other variables are urbanand non agricultural land use, woodland area, terrain, and economic activity. These variables maybe characterized as being either “limiting variables” or “related variables”. We will tackle thelimiting variables first.

The limiting variablesIn some degree limiting variables restrict the possible occurrence of cropland. That is, a certainpercentage of a limiting variable occurring in an area will set an absolute upper limit on thepercentage of the mapped variable (cropland) that can occur in the same area. Two limitingvariables are being employed in this exercise – urban landuse and woodland. An area devoted tourban landuse precludes the occurrence of cropland. In a categorical fashion, this may be expressedby the following rule: if urban landuses are present, then there can be no cropland. If there are nourban landuses, then cropland can exist (see Figure 1). In terms of the six categories of cropland,note that the presence of urban land use restricts an area to cropland category 1, namely, 0%cropland. All six classes are possible where no urban land use occurs.

cropland classes (%)1 2 3 4 5 6

urban land use 0-9 10-29 30-49 50-69 70-89 90-100

presentnot present

cropland not possible

Figure 1: the limiting variable urban landuse

It is also possible to compute a precise adjusted percentage for the cropland density, using aformula first published by J.K. Wright. It is called the computation of fractional parts ofdensities. In the formula below, D stands for Density, A stands for Area:

D

D D A

Ano m m

m

=−

−=

− ∗

−= =

1

52 0 0 1

1 0 1

52

0 957 7

( . )

. ..

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Thus, for a county with 52% cropland, 10% of the county covered by urban landuse, the percentageof cropland in the remaining county area must be approximately 58%.

TASK 1Revise estimates of county cropland density using this formula and the layer of urbanized areas.The first step is to generalize this layer, and the second is to do the computations.

ToolsYou will be working with ArcView’s Spatial Analyst, a raster-based GIS module.Open the view called Map Calculations by making the dasy.apr window active, selecting theview icon and double clicking on the label Map Calculations. We will store all the followingcalculations in this view.

Step 1 – Generalizing the urbanized layerLook at the towns theme in the Cropland in Ohio 1959 view. Notice there are many regionsthat are only a pixel or a couple of pixels in size, at this level of resolution. To include these willgenerate an overly complex visual display, partially obscuring the dasymetric patterns. So beforewe begin to compute the fractional densities, you need to eliminate the smallest regions from thismap, creating a new, generalized towns map. Here’s how to eliminate the regions.

• Select towns layer• Go to Analysis: Neighborhood Statistics...• Statistics: Sum• Neighborhood: Rectangle• 3x3 Cells

The result map has classes in somewhat concentric bands. You can generalize your towns map bydeciding on which of the bands to keep and which ones to get rid of by setting them to No Data.Do this first graphically. Double click the theme’s legend and set the first couple of classes towhite, and the remaining to any identical shade. Look at the generalization effect. Once you havedecided how much to generalize create a new layer with the new classification.

• Go to Reclassify...• Click into New Values• Make the two classes• Keep the No Data class and reassign No Data to the classes you want to get rid of• Assign 1 to the remaining classes.

To rename the towns layer go to THEME: PROPERTIES and label the new towns layer newtowns. Cut the theme from the Cropland in Ohio view and paste it into the Map Calculationsview (in the EDIT menu, CUT & PASTE). You just created a map layer whose cells containfrequency counts of the number of non-“No Data” cells around each pixel. A 3x3 cell selection isthe diagonal size (in pixels) of the square moving search window. Compare new towns to townsand you’ll see how much simplification you have effected. Write down in your journal log theexact parameter values you use.

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Step 2 – Computing fractional parts of densities.In any GIS, numerical analysis can be somewhat convoluted – ArcView is no different. The

following steps will build up the various parts of J. K. Wright’s original formula given above.Spatial Analyst operators will be used to apply it as a whole to the choropleth map, and the resultwill be a new map layer of revised cropland density values. Here’s the tricky part – ArcViewwants everything in integers (no percentages) to retain access to the attribute tables of raster layers(in ArcView lingo called GRIDS). We need to transform J. K. Wright’s original formulaalgebraically, to express it in terms GIS raster layers.

revised crop density =

original crop density - (0 urban area [%])

1 - urban area [%]

∗

revised crop density = original crop density

county area in pixels

county area in pixels

urban area in pixels

urban area in pixels−

revised crop density =

original crop density county area in pixels

nonurban area in pixels

∗

First, you will compute a frequency count of how many cells are contained in each zone of yourchoropleth map (‘county areas in pixels’ in above revised formula). Make the Cropland in Ohioview active. Go to ANALYSIS: MAP CALCULATOR. To see the complete variable names inthe left list, make the Map Calculator Window bigger by dragging its lower right corner towardsthe lower right. Double click on the [choropleth crop.Count] layer in the Layers List. Click onthe evaluate button to compute a new layer. Rename the new theme called “Map Calculation 1” tocounty areas and cut/paste this map into the Map Calculations view. You just computed thevariable ‘county areas in pixels’ from above formula).

Before we can calculate the area devoted to urban landuse within each county we need tocombine the urban layer with the county layer. Copy the choropleth % crop into the MapCalculations view. Select new towns and go to ANALYSIS: RECLASSIFY. The categorywith value No Data has to be reassigned to value 1 (one). The category with 1 is reassigned 0(zero). Make both this reclassified towns map and choropleth % crop active by shift clicking ontheir names in the legend. Go to DASYMETRIC: COMBINE. This command is a script that Iwrote in Avenue (ArcView’s programming language) that generates unique values for all existingcombinations of values in the two input themes. The result is a new theme called “Combined X w/Y”, where X and Y are the input theme names. Label the new theme choropleth w/urban. Youcan explore all the scripts used for this lab by making the ohio.apr window active, selecting thescript icon from the list and double clicking on any of the labels.

Next, we need the denominator of above formula, the nonurban area. We will compute this inseveral steps. First, we computes a frequency count of urban landuse pixels within each county ofthe “choropleth % crop” map. So, go to ANALYSIS: MAP CALCULATOR again, and doubleclick on the [choropleth w/urban.Count] layer in the Layers List. Click on the evaluate buttonto compute a new layer. Rename the new theme called “Map Calculation 1” to area of chorow/urban and cut/paste this map into the Map Calculations view. Ok, now we need to cookie

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cut the urban areas out of this map to get the remaining area that is not devoted to urban landuse.The new towns theme is our cookie cutter mask. To do this we need to reclassify the towns layerfirst. Select new towns and go to ANALYSIS: RECLASSIFY. The category with value 1(urban) needs to be reclassified to No Data, and category No Data is reassigned value 1 (one).Next, make the reclassed of new towns (cookie cutter) and the area of choro w/urban themesactive (shift click) and go to DASYMETRIC: COVER. Rename the result theme nonurbanarea. Inspecting again above formula you have now computed the denominator of J.K. Wright’sformula, sort of (e.g. ‘nonurban area in pixels’). This map contains the number of pixels withineach county that that IS NOT devoted to urban uses. Meaning, the theme nonurban area givesyou the area of each county that could be devoted to cropland.

So, now we have got all those things we need for above formula, the original crop density(choropleth % crop), the county areas, and the nonurban areas. The original choropleth map takes abit of manual reclassifying, modify the choropleth % crop theme in the Map Calculations viewand change each class value to correspond with the actual cropland density value. Since onlyintegers are acceptable, you can multiply by 10, thus class 1 becomes 111. Class 2 becomes 124.Put a note into the journal – you will have to reclassify other map layers accordingly. You canround the values if you wish. Make the choropleth % crop theme in the Map Calculations viewactive, go to ANALYSIS: RECLASSIFY and reclassify each class accordingly.

Okay (phew!), now we can compute the fractional parts of densities. Compute below formulawith the Map Calculator:

([Reclass of choropleth % crop] * [county area]) / [nonurban area])

Label the new theme choropleth w/urban limits, put a hardcopy into the journal.Remember, the values that you see in this map have to be divided by 10 or 100 to reflect thereclassification you performed earlier. Therefore a value of 921 is 92,1 percent (if areclassification factor of 10 was used).

Now compare this map to the original choropleth % crop. Which counties have drasticallyrevised cropland values? Why do you think it is so, for each drastically changed county? Include ahard copy of this map in the journal file, annotate the changed counties and write a note on the maptelling why (presence of big city, presence of lots of mines, etc.).

To summarize the previous steps: we have taken an initial pass at revising the choroplethestimates with a first limiting variable ‘urban areas’. The choropleth w/urban limits theme isthe map layer to use for all remaining tasks involving crop density overlays.

TASK 2 – second limiting variable woodlandThe second limiting variable we will introduce into our dasymetric model is percentage area inwoodland. Adding its six categories (0-9%, 10-29%, etc.) to the above classification systemexpands the number of potential categories from 12 to 72, as shown in Figure 2.

cropland classes (%)1 2 3 4 5 6

urban land use woodland 0-9 10-29 30-49 50-69 70-89 90-100

1 0-92 10-293 30-494 50-69present

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5 70-896 90-1001 0-92 10-293 30-494 50-695 70-896 90-100

cropland not possible

Figure 2: the limiting variable woodland

But where urban types are used, we are restricted to class 1 (from Figure 1, this follows directly).Thus 30 of the 72 categories are immediately eliminated (shown by the gray shading in thediagram). The addition of woodland will eliminate more, according to the following rule.Given a certain percentage of woodland in an area, only the balance of the area may be incropland. Thus if 35% of an area is in woodland, then the highest possible value of croplandwhich can exist anywhere else in the county is 65%. Notice that some portions could have as muchas 95% cropland, if these were offset by other portions of lesser percentage cropland, to the extentthat the sum of all cropland percentages in the county as a whole average out to the enumeratedtotal. So as shown in Figure 2, if an area is 90% woodland, then only 10% of that area may be incropland. If it is 70 -89%, then only classes (1) or (2) are possible, etc. Through this directprocess of limiting variables, we can eliminate 15 more categories for the map.

This could be accomplished using J.K. Wright’s formula, above, and repeating the process inturn for each of the six woodland classes, subsequently covering each class with the others. Thiswould in the end provide 28 revised values for each county’s cropland density. However, let’s berealistic about time and do it in classed fashion. This way we will get a prediction in six classes,and then approximate the unclassed solution by map overlay.Make a duplicate of the woodland theme in the Cropland in Ohio 1959 view and paste it intothe Map Calculations view. Reclassify this theme. If you multiplied earlier by 10, your newclass labels should be 99, 299, 499, etc. Next, click on the legend text to invert ranges fromwoodland to cropland, as follows:

class 36 is assigned class 99 new legend text should read: 0 - 9 % cropclass 35 is assigned class 299 new legend text should read: 10 - 29 % cropclass 34 is assigned class 499 new legend text should read: 30 - 49 % cropclass 33 is assigned class 699 new legend text should read: 50 - 69 % cropclass 32 is assigned class 899 new legend text should read: 70 - 89 % cropclass 31 is assigned class 999 new legend text should read: 90 -100 % crop

Rename this theme Reclass of woodland % crop and change the legend text in this theme tomatch the above classes. Use the Legend Editor and fill in the label field accordingly.

Now overlay to produce the limiting variables map, and then overlay that on the unclassedchoropleth to make your first refined estimate of cropland densities. The second formulaapproximates what Wright’s density formula would have produced by taking the minimum valuecell-by-cell throughout the map matrices. Here is how: Use the Map Calculator and type in

notpresent

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([Reclass of new towns] * [Reclass of woodland % crop])

Rename the result limiting variables. Now select both, limiting variables and choroplethw/urban limits in your Map Calculations view by shift clicking on the theme names. Go toANALYSIS: CELL STATISTICS… and choose MINIMUM from the drop down list.Rename the result choropleth w/ limits.

Clean up the legend and color scheme. Put hardcopies into the journal file, annotating withnotes about what patterns you see. Put a statement in the journal file comparing choropleth w/limits with choropleth w/urban limits, and with choropleth % crop (original choropleth map).

TASK 3 – Related variablesRelated variables may be associated with the variable being mapped in complex ways. In thiscase the related variables are the terrain and the economy map layers. Figure 4 shows the croplandclasses (99, 299, 499, etc.) associated with each combination of the related variables’ classes.

economyterrain 21 22 23 24 25 26 27 28 29

steep 44hilly 43

rolling 42level 44

cropland classes99 699299 899499 999

21 Truck Farming (w/ dairy & poultry) 26 Dairy Farming (w/ large areas in crops)22 Truck Farming (w/ mixed or cash grain farming) 27 General Farming23 Cash-grain Farming (w/ some livestock) 28 Dairy Farming (w/ little cropland)24 Mixed Farming (primarily cash crops) 29 Livestock Ranching25 Mixed Farming

Figure 3: cropland classes for related variables

Accomplishing this step is quite similar to the woodland task. There are two ways to do it. Thefirst is to overlay the two layers with the COMBINE command and then manually reclassifying abunch of classes. This is pretty inefficient, will take a long time and probably result in numerousmistakes. The easier way is to use the CROSS command in the DASYMETRIC menu. CROSSis a script that combines two map layers (terrain & economy) and automatically reclassifies theoutput theme based on conditions shown in Figure 4. Make the Cropland Ohio view active.Select CROSS and a new theme will be added to the view called related variables. Put ahardcopy in your journal. Cut/Paste this theme into the Map Calculations view.

Compare this with the choropleth w/ limits in your journal notes. Notice that neither thelimiting variables nor the related variables map should display county boundaries – this isbecause neither have been derived from county-based enumeration, and therefore including theboundaries might mislead the map reader about data sources. As you combine these maps withthe original choropleth data (e.g., with choropleth w/urban limits), the boundaries will be

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incorporated automatically. Think about why that distinction might be important for decisionmaking, in your journal.

Step 2 – Dasymetric modelWith the related variables and the limiting variables layers you have made so far, you can predictwhat the dasymetric map should look like, in the absence of any choropleth data. The procedureassumes that the related variables map layer is a good basic estimator of variations in croplanddensity that actually existed in 1959. The cartographer feels confident that if surface topographyand economic activity were the only data available, then the “related variables” map would byitself yield a pretty good approximation of the spatial distribution of cropland. The purposeserved by the limiting variables is to limit the estimates at any place. The general rule is that therelated variables estimate is valid except where the limiting variables (i.e., maximum possiblecropland) are lower than the estimate. Figure 5 illustrates the principle. If the predicted class inthe related variables map layer is category 5 (that is, 70 - 89% cropland) but there is an upperlimit of category 2 (that is, 10 - 29% cropland) for this area, then this area is limited to class 2.However, if the limited variables map layer shows an upper limit of category 6, as in the right halfof the limits map in the Figure, then the predicted cropland class (class 4) is accepted as valid forthe area.

Figure 4: dasymetric overlay principle

Statistically speaking, your dasymetric solution displays the more conservative of the twoestimates for percentage cropland. This is the estimate of cropland based solely on limiting andrelated variables.

Select both, limiting variables and related variables in your Map Calculations view by shiftclicking on the theme names. Go to ANALYSIS: CELL STATISTICS… and selectMINIMUM from the drop down list. Rename the result dasymetric model. Color code, labellegend appropriately, and put a hardcopy into the journal file.

Dasymetric mapIt is also possible to refine the original choropleth data, to see the dasymetric pattern. This is theestimate of cropland based on related and limiting variables, AND incorporating the originalchoropleth data. Repeat the MINIMUM function, this time by selecting related variables andchoropleth w/ limits in your Map Calculations view. Rename the new map dasymetric map.

Color code, label legend appropriately, and put a hardcopy into the journal file. Once youhave the symbology in comparable visual form, compare the dasymetric model and thedasymetric map with each other, and with the original choropleth map (choropleth % crop). Whatdifferences do you see – where are the high and low regions, where is the pattern uniform, non-

2 6

limiting dasymetric

2

5

4

related

5

4+ =

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uniform, etc.? To compare your result maps with the ‘correct’ ones, you can have a peek at themaps attached at the end of this handout.

Things to consider when looking at the result maps...How would you appropriately color your maps? Did your final displays turn out as youintended?

Think about this – the “choropleth % crop” and the “dasymetric model” are bothpredictive models of cropland density in Ohio in 1959. What is the difference between thetwo types of classification procedures (generalization), and the two resulting maps? Next,consider the “dasymetric map” which is a refinement of the original cropland densityestimate. What do you gain by combining the original choropleth estimate with thedasymetric model?

Now think in terms of the geography, what WAS the pattern of cropland in Ohio in 1959?How much of this pattern appears to be associated (or even determined by) the limiting andrelated variables? Is it uniform across the study area? Where does it vary, and why? Try tothink about it as a geographer – elevate your mind.

© Original by Dr. B.P. Buttenfield, ported to ArcView and modified by sara fabrikant, 2001.

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