The analytical power of any GIS system lies in its ability to integrate, transform, analyze, and model data using simple to complex mathematical functions and operations. In the lesson this week, we will look at the fundamentals of surface analysis and how a flexible map algebra raster analysis environment can be used for the analysis of field data.

Figure 7.0: Applying different map algebra methods for performing a modeling analysis.

Click for a text description of the Lesson 7 analysis summary image.

This image shows examples of modelling steps involved in the Lesson 7 analysis, including creating a Euclidean Distance analysis, creating a weighted roads raster layer, estimating the population of school-aged children in a district, and mapping that population to see its spatial distribution, and finally doing a weighted-roads cost allocation distance analysis.

Credit: Blanford, © Penn State University, is licensed under CC BY-NC-SA 4.0

Learning Outcomes

At the successful completion of Lesson 7, you should be able to:

• describe data models for field data: regular grid, triangulated irregular network, closed form mathematical function, control points; and discuss how the choice of model may affect subsequent analysis;
• explain the map algebra concept and describe focal operations, local operations, and between-map operations;
• understand the idea of slope and aspect as a vector field;
• explain how slope or gradient can be determined from a grid of height values;
• describe how surface aspect may be derived from a grid of height values;
• re-express these operations as local operations in map algebra;
• describe how map algebra operations can be combined to develop complex functionality.

Checklist

Lesson 7 is one week in length. (See the Calendar in Canvas for specific due dates.) The following items must be completed by the end of the week. You may find it useful to print this page out first so that you can follow along with the directions.

Questions?

Please use the 'Discussion - Week 7' forum to ask for clarification on any of these concepts and ideas. Hopefully, some of your classmates will be able to help with answering your questions, and I will also provide further commentary there where appropriate.

Once you have field data, whether as a result of interpolation, or based on more complete original data, perhaps from aerial surveys or remotely sensed imagery, you will likely want to analyze it in a variety of ways. Use the outputs of the analysis to make an informed decision. Over the past few weeks, you have been using many spatial analysis functions (review Table 2.0, Lesson 2). This week, we will look at how we can integrate outputs that may have been created from these different functions to help us make an informed decision.

Map algebra is a framework for thinking about analytical operations applied to field data. It is most readily understood in the case of field data that are stored as a grid of values but is, in principle, applicable to any type of field data.

The map algebra framework was devised by Dana Tomlin and is presented in his 1990 book Geographical Information Systems and Cartographic Modeling (Prentice Hall: Englewood Cliffs, NJ), which you should consult for a more detailed treatment than is given here. Another good reference on map algebra (and much else besides) is GIS Modeling in Raster (Wiley: New York, 2001) by Michael DeMers.

Many GIS (including Esri’s ArcGIS) support map algebra. In ArcGIS, the tool most closely related to map algebra is called the ‘raster calculator'.

Basic concepts

The fundamental concepts in map algebra are the same as those in mathematical algebra, that is:

• Values are the 'things' on which the algebra operates. Input data and output data (results) are presented as grids of values. Values can be categorical (nominal or ordinal) or numerical.
• Operators may be applied to single values to transform them, or between two or more values to produce a new value. In mathematical algebra, the minus sign '-' is an operator that negates a single value when placed in front of it, as in -5. The plus sign '+' is also an operator, signifying the addition operation, which, when applied between two values, produces a new value: 1 + 2 = 3
• Functions are more complex, but still well defined, operations that produce a new output value from a set of input values. The input set may be a single value, as in $lo{g}_{10}\left( \left\{100\right\} \right)=2$, or a set of values, as in $\text{average}\left( \{1,2,3,4\} \right)=2.5$.

To apply an operation or function to these values, there are many different ways to proceed that range in complexity from simple to advanced and from local to global operations.

Below are examples of local, focal, zonal, and global operations and how they work when evaluating cells. Each of the operation types differs in how much of the cell’s neighborhood is used by the operator or in the operation.

Table 7.0

Local	A local operation or function in map algebra is applied to each individual cell value in isolation.	Figure 7.1. Result of a local operation/function Credit: Blanford, © Penn State University, licensed under CC BY-NC-SA 4.0	This means that the value at each location in the output grid is arrived at by evaluating only values at the location of each individual cell.
Focal	Uses a user-defined neighborhood surrounding a cell (often 3 x 3 but other neighborhood sizes and shapes are also possible) in the map algebra operation.	Figure 7.2 Result of the focal max. The result at each location in the output grid is determined by combining values focused at the corresponding location in the input grid or grids, as shown by the shading.  Credit: Blanford, © Penn State University, licensed under CC BY-NC-SA 4.0	Many functions can be applied focally in this way, such as maximum, minimum, mean (or average), median, standard deviation, and so on. A different choice of focal neighborhood will alter the output grid that results when a focal function is applied. There is no requirement that the neighborhood be symmetrical about the focal grid cell, as shown in this example. However, it is consistent in size and shape when it is applied to each cell in the data set. A non-symmetrical neighborhood might have application in understanding how air pollution spreads given a prevailing wind direction.
Zonal	Is applied to a set of map zones (e.g., counties, Congressional Districts, etc.).	Figure 7.3 Example of a zonal operation where the result at each location in the output grid is determined from the set of values in the zone that the cell falls within as shown by the shading. Credit: Blanford, © Penn State University, licensed under CC BY-NC-SA 4.0	Zonal operations and functions are an extension of the focal concept. Rather than define operations with respect to each grid cell, a set of map zones is defined (for example, counties), and operations or functions are applied with respect to these zones.
Global	Values at each grid cell in an output grid may potentially depend on the values at all grid cell locations in the input grid(s).	Figure 7.4 Example of a global operation where the result at each location in the output grid is determined from the set of values at that location and other locations in the input grid, as shown by the shading. Credit: Blanford, © Penn State University, licensed under CC BY-NC-SA 4.0	Global operations and functions include operations that find the cost (in time or money) of the shortest path from a specified location (e.g., the yellow cell in the figure) to every other location. Such operations may have to take into account values at all locations in a grid to find the correct answer.

The measures discussed in this section are just a small sample of the types of surface analysis measures that can be devised. Map algebra operations can vary in complexity from simple to advanced.

In doing so, they may integrate single mathematical functions or increase in complexity by integrating multi-step mathematical functions/operations that may be evaluative (e.g., nested functions) and/or dynamical in nature (e.g., spatio-temporal models, agent-based models, process-based models, overlay models (more on this in Lesson 8), depending on what is being analyzed.

Agent-based models are spatio-temporal models that are often applied in cell-based environments and contain agents that move across the landscape reacting to the environment based on a set of pre-defined rules.

Process-based models might be used to predict the temporal fluctuations in soil moisture, water levels and hydrologic networks or for disease prediction where temporal fluctuations in temperature and rainfall might affect the host-pathogen interaction and disease outcome in the environment (e.g. Blanford, et al. 2013 Scientific Reports).

The examples below should give you a feel for the flexibility of the map algebra framework and how it can be used to capture simple processes as well as more complex processes.

In this week's project, you will have an opportunity to explore map algebra more thoroughly in a practical setting.

So far in this course, we have only considered attribute data types that are single-valued, whether that value is categorical or numerical. In spatial analysis, we frequently encounter attributes that are not conveniently represented in this way. In particular, we may need to use vectors to represent some types of data.

A vector is a quantity that has both value (or magnitude) and direction. The most obvious vector in real-life applications is wind, which has a speed (its magnitude or strength) and direction. Without wind direction information, wind speed information is not very useful in many applications. For example, an aircraft navigator needs to know both wind speed and direction to accurately plot a course and to estimate arrival times or fuel requirements.

The most fundamental vector field is the gradient field associated with any scalar (i.e., simple numerical) field. This often has practical applications. For example, the gradient field of atmospheric pressure is important in meteorology in determining the path of storm systems and wind directions.

Background

Now, let's continue our work on data from Central Pennsylvania, where Penn State's University Park campus is located. This week, we'll see how this ancient topography affects the contemporary problem of determining potential locations for a new high school.

Note that this project is more involved than some of the earlier ones. The instructions are also less comprehensive than in previous projects - by now you should be getting used to finding your way around in ArcGIS. Of course, if you are struggling, as always, you should post questions to the discussion forums.

Introduction

The Centre County region of Pennsylvania is among the fastest growing regions in the state, largely as a result of the presence of Penn State in the largest town, State College. Growth is putting pressure on many of the region's resources, and some thought is currently being given to the provision of high schools in the region. In this project, we will use raster analysis based on road transport in the region to determine potential sites for a new school. This will demonstrate how complex analysis tasks can be performed by combining results from a series of relatively simple analysis steps.

Project Resources

The data files you need for Project 7 are available in Canvas. If you have any difficulty downloading this file, please contact me.

Once you have downloaded the file, double-click on the Lesson7_project.ppkx file to open it in ArcGIS Pro or unzip the .zip file for use in ArcGIS Desktop 10.X.

This project contains a geodatabase with the lesson data (Lesson7.gdb) and a raster file with a digital elevation model to provide context.

Open the ArcGIS Pro project file or ArcGIS Desktop .mxd file to find layers as follows:

• highSchools - four high schools in Centre County, Pennsylvania
• centreCountySchoolDistricts - the corresponding school districts
• majorRoads - major roads in the region
• localRoads - minor roads in the region
• centreCountyCivilDivisions - showing townships and boroughs in the county
• representativePopInSchoolDistricts - a point layer derived from census block group centroids with attributes for total population and children aged between 5 and 17 from the 2017 release of the 5-year American Community Survey population estimates
• centreBGdemographics - more complete demographic data from the 2017 release of the 5-year American Community Survey population estimates associated with polygons representing the block groups for which the data were collected
• centredem500 - the topography of the county at 500 meter resolution.

Summary of Project 7 Deliverables

For Project 7, the minimum items you are required to have in your write-up are:

• Describe in your write-up how the distance analysis operation works, including commentary on how you would combine multiple distance analysis results (one for each high school) to produce an allocation analysis output and the differences in the straight line distance allocation and the actual allocation of places to school districts in the present example.
• Describe in your write-up how you created the roads raster.
• Perform the cost weighted distance analysis for high schools using the roads raster layer. Examine the resulting allocation layer. How does it differ from the straight-line distance allocation result? Do the roads account for all the inconsistencies between the straight line distance allocation and the actual school districts? Answer these questions in your write-up.
• Estimate the number of school-aged children in the four school districts, and also in the road travel cost-weighted distance allocation zones associated with each school, and put these estimates in your write-up. Also, describe how you arrived at your estimates.
• Insert into your write-up a map and other details of your proposals for a new high school and associated district, including arguments for and against, possible problems with your analysis, maps, and explanations of any analysis carried out.

Questions?

Please use the 'Discussion - Lesson 7' forum to ask for clarification on any of these concepts and ideas. Hopefully, some of your classmates will be able to help with answering your questions, and I will also provide further commentary there where appropriate.

Figure 7.5: Setting Analysis Environments.

Click for a text description of the Setting Analysis Environments image.

Both the Mask and Processing Extent should be set to centreCountyCivilDivisions in the Analysis Environment window.

Credit: A. Griffin © Penn State is licensed under CC BY-NC-SA 4.0

The first analysis we will do uses the Spatial Analyst Tools - Distance - Euclidean Allocation tool from the Tools menu (Figure 7.6). This allocates each part of the map to the closest one of a set of points and is the raster equivalent of proximity polygons.

Figure 7.6: The Euclidean Allocation user interface with parameters set to generate an allocation of areas to high schools.

Click for a text description of the Euclidean Allocation user interface image.

The input features parameter should be set to the high schools layer. The source field should be set to ID. Name the output allocation layer something sensible, such as Euclidean_Allocation_Schools. Be sure that you save your output layer to the gdb and not another folder. If one of the schools has no cells allocated to it, you've likely saved the output outside the gdb. The cell size will be autopopulated with the value you set for cell size in the Analysis Environments settings dialogue.

Credit: A. Griffin © Penn State is licensed under CC BY-NC-SA 4.0

Running this Euclidean Allocation (straight line distance) analysis will produce an allocation layer (Figure 7.7).

Figure 7.7: Allocation layer created from the Euclidean Allocation analysis.

Click for a text description of the Euclidean Allocation analysis output image.

This image shows the output of the Euclidean Allocation analysis with the settings described in the lesson. There is a mismatch in alignment between the allocation areas generated by the tool and the actual school districts. The allocation districts are most poorly aligned in areas that contain mountain ridges.

Credit: A. Griffin © Penn State is licensed under CC BY-NC-SA 4.0

You can also request a distance layer (Figure 7.8).

Figure 7.8: Results of the Euclidean Distance analysis for high schools.

Click for a text description of the Euclidean Distance analysis outputs image.

In the Euclidean Distance analysis output, the raster cells contain values that describe the distance to the nearest school. The image looks like concentric buffer rings whose overlaps have been dissolved.

Credit: A. Griffin © Penn State is licensed under CC BY-NC-SA 4.0

You can further analyze these layers. For example, it may be easier to read the distance analysis if you create contour lines. The results of the allocation analysis can be converted to vector polygons, which might make subsequent analysis operations easier to perform, depending on which approach you decide to take.

Clearly, roads are a major factor in the difference between straight-line distance allocation of school districts and the actual school districts. In this part of the project, you will create a raster layer representing the roads of Centre County to use in a second, roads-based distance analysis.

Creating a roads raster layer using map algebra operations

1. Using the Conversion Tools - To Raster - Polyline to Raster... tool, make raster layers from the majorRoads and localRoads layers. Make sure that you use the same resolution for both layers. Consider what spatial resolution might be appropriate for this analysis.
2. Use the Spatial Analyst Tools - Reclass - Reclassify and Spatial Analyst Tools - Map Algebra - Raster Calculator tools to manipulate and combine these layers into a single roads layer where major road cells have value 1, minor road cells have value 2, and off-road cells have some high value (say 100).

The end result will look something like this (although you should consider creating a better designed map):

Figure 7.9: Part of a combined roads raster layer. Red cells (major roads) have a value of 1, orange cells (minor roads) have value of 2, and clear cells (that is, background color; all non-road cells) have value of 100.

Credit: A. Griffin © Penn State is licensed under CC BY-NC-SA 4.0

Using the roads raster layer for distance analysis

Distance analysis using a roads raster layer is straightforward. The values in the layer are regarded as 'weights' indicating how much more expensive it is to traverse that cell than if the cell were unweighted. Thus, with the roads layer you just created, traveling on major roads incurs no penalty, traveling on local roads is twice as expensive (takes twice as long), and traveling off-road is very slow indeed (100 times slower). These are not accurately determined weights, but serve to demonstrate the potential of these methods.

Select the Spatial Analyst Tools - Distance - Cost Allocation tool (Figure 7.10).

Here you are performing another distance analysis, but this time weighted by the roads raster layer you just created. Again, request an Allocation output from the analysis. (You might also be interested to request the Output distance raster to look at the distances that accumulate for each cell.)

Figure 7.10: The Cost Allocation user interface for doing cost distance analysis. Note that you will probably have named your input cost allocation raster something different. This is simply the final, weighted roads layer you produced. Also notice that there are several parameters you can specify to further tweak the cost calculations. For now, we’ll not use these options, but if you have the time, you could experiment with them and consider which might be relevant to this particular application of cost distance analysis. Hover over the information icon that appears when you mouse over the parameter name for a more detailed explanation of what the parameter manipulates.

Click for a text description of the Cost Allocation tool image.

Again, your input feature layer is the high schools layer and the source field is ID. Your input cost raster is the roads layer you created in the prior step of the exercise. Give your output a sensible name such as Road_Cost_Allocation_Schools.

Credit: A. Griffin © Penn State is licensed under CC BY-NC-SA 4.0

Estimating the school-aged population in each of your new school districts

Given the representativePopInSchoolDistricts and CentreCountyBGDemographics layers, the next task is to estimate how many children are in each school district, as a preliminary step before deciding where a new school would be most useful.

There are many different ways that you could tackle this task.

Some of them involve raster operations (recall the approaches used in the Texas redistricting example in Lesson 3), and the Spatial Analyst Tools - Zonal - Zonal Statistics tool. But before you can use these, you would have to construct a raster representation of the school-aged population. Again, consider how this was done in the Texas example.

A more straightforward approach will use Analysis - Summarize Within (Figure 7.11), a Spatial Join (Analysis - Spatial Join, Figure 7.11), or Analysis Tools - Overlay - Spatial Join from the Tools menu.

Figure 7.11: Commonly needed analysis tools can be accessed from the quick-start menu in the Analysis ribbon tab. Access the full list of quick-start tools by clicking the bottom arrow at the right-hand side of the icon group.

Click for a text description of the Quick Start Tools Menu image.

This image shows the quick start tools menu. It organizes commonly used tools by function. Tools to implement several of the approaches described above for estimating population can be found in this menu.

Credit: A. Griffin © Penn State is licensed under CC BY-NC-SA 4.0

In analysis like this, there is no 'right' or 'wrong' way, just what works, so here's the deliverable - you figure out how to get there. Do this step for both the original districts and the road-cost-weighted districts.

And so... where to put a new school, and what should be its district?!

The headline above summarizes the last part of the project. Based on the analyses already carried out, and any other analyses required to support your answer, locate a new high school in Centre County.

As a minimum, you should use the editing tools (Edit – Create, Figure 7.12) to add a school to the highSchools layer and create a map of the new school and what the new school's associated district would be. You should insert this map, together with a description of how you arrived at it, into your Project 7 write-up. Note that school districts are always made up of a contiguous collection of townships and/or boroughs, and that the centreCountyCivilDivisions layer shows these.

Figure 7.12: Add new features to a feature class through the Create Features pane for the layer.

Credit: A. Griffin © Penn State is licensed under CC BY-NC-SA 4.0

Please put your write-up, or a link to your write-up, in the Project 7 drop-box.

Summary of Project 7 Deliverables

For Project 7, the items you are required to have in your write-up are:

• Describe in your write-up how the distance analysis operation works, including commentary on how you would combine multiple distance analysis results (one for each high school) to produce an allocation analysis output and the differences in the straight-line distance allocation and the actual allocation of places to school districts in the present example. In your write-up, be sure to include maps and associated tables you feel will help illustrate this part of your analysis.
• Describe in your write-up how you created the roads raster. In your write-up, be sure to include maps and associated tables you feel will help illustrate this part of your analysis.
• Perform the cost-weighted distance analysis for the high schools using the roads raster layer. Examine the resulting allocation layer. How does it differ from the straight-line distance allocation result? Do the roads account for all the inconsistencies between the straight-line distance allocation and the actual school districts? Answer these questions in your write-up. In your write-up, be sure to include maps and associated tables you feel will help illustrate this part of your analysis.
• Estimate the number of school-aged children in the four school districts and put these estimates in your write-up. Also, describe how you arrived at your estimates. In your write-up, be sure to include maps and associated tables you feel will help illustrate this part of your analysis.
• Insert into your Project 7 write-up a map and other details of your proposal for a new high school and associated district, including an estimate of the number of school-aged children in each zone, arguments for and against the particular location and district boundaries you are proposing, possible problems with your analysis, maps, and explanations of any analysis carried out to support your decision-making. You should also calculate the road travel cost-weighted distance allocation zone associated with each school and show your results in a map (or maps).

I suggest that you review the Lesson 7 Overview page to be sure you have completed the all required work for Lesson 7.

That's it for Project 7!

There is no specific output required this week, but you should be aiming to make some progress on your project this week.

Additional details about the project can be found on the Term Project Overview Page.

Questions?

Please use the Discussion - General Questions and Technical Help discussion forum to ask any questions now or at any point during this project.

Lesson 7 Deliverables

1. Complete the Lesson 7 quiz.
2. Complete the Lesson 7 Weekly Project activities. This includes inserting maps and graphics into your write-up along with accompanying commentary. Submit your assignment to the 'Project 7 Drop Box' provided in Lesson 7 in Canvas.

Reminder - Complete all of the Lesson 7 tasks!

You have reached the end of Lesson 7! Double-check the to-do list on the Lesson 7 Overview page to make sure you have completed all of the activities listed there before you begin Lesson 8.

Additional Resources

Least-cost paths are used for more than modeling the movements of people or goods. They are frequently used in animal ecology as well.

A number of researchers have turned to least-cost paths to help in identifying potential movement corridors that should be preserved to maintain landscape connectivity and avoid negative effects on animal population numbers and density because of increasing habitat fragmentation.

A key challenge in this type of modeling lies in defining the habitat and animal species characteristics that can inform the cost weights that describe how the animal moves through the landscape.

If these applications are of interest to you, the two papers below are worth looking at: