The links below provide an outline of the material for this lesson. Be sure to carefully read through the entire lesson before returning to Canvas to submit your assignments.
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.
At the successful completion of Lesson 7, you should be able to:
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.
Step | Activity | Access/Directions |
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1 | Work through Lesson 7 | You are in Lesson 7 online content now. You are on the Overview page right now. |
2 | Reading Assignment | You need to read the following selections:
NOTE: This reading is available through the Penn State Library’s e-reserves. You may access these files directly through the library Resources link in Canvas. Once you click on the Library Resources tab, you will select the link titled “E-Reserves for GEOG 586 - Geographical Information Analysis." A list of available materials for this course will be displayed. Select the appropriate file from the list to complete the reading assignment. The required course text does not cover all the material we need, so there is some information in the commentaries for this lesson that is not covered at all in the textbook reading assignments. In particular, read carefully the online information for this lesson on "Map Algebra" and "Terrain Analysis." After you've completed the reading, get back online and supplement your reading from the commentary material, then test your knowledge with the self-test quiz. |
3 | Weekly Assignment | This week, you will apply various surface analysis methods and spatial analysis functions, including more complex map algebra operations, to choose a suitable location for a new high school using ArcGIS. (The materials for Project 7 can be found on Canvas.) |
4 | Term Project | There is no specific output required this week, but you should be aiming to make some progress on your project this week. |
5 | Lesson 7 Deliverables |
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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 and 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 [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.
Before we go any further, you need to read the following text, which is available through the Library Resources tab in Canvas:
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'.
The fundamental concepts in map algebra are the same as those in mathematical algebra, that is:
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.
Local | A local operation or function in map algebra is applied to each individual cell value in isolation. |
Credit: Blanford, © Penn State University, licensed under CC BY-NC-SA 4.0 [1]
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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. |
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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. |
Credit: Blanford, © Penn State University, licensed under CC BY-NC-SA 4.0 [1]
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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. |
Zonal | Is applied to a set of map zones (e.g., counties, Congressional Districts, etc.). |
Credit: Blanford, © Penn State University, licensed under CC BY-NC-SA 4.0 [1]
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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). |
Credit: Blanford, © Penn State University, licensed under CC BY-NC-SA 4.0 [1]
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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. |
Before we go any further, you need to read the following text, which is available through the Library Resources tab in Canvas:
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 [3]).
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.
Relative relief | Relative relief, from the definition in the text, is readily expressed as a map algebra function: rel_relief = focal_max( [elevation] ) – focal min( [elevation] ) where the focal region is defined accordingly. |
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Surface gradient and aspect | Surface gradient is more complex to model, requiring a number of steps. First, two focal functions must be defined to calculate the slope in two orthogonal directions. These will be similar functions but must have specially defined focal areas that pick out the immediately adjacent grid cells on either side of the focal cell in each of the two cardinal directions. If these resulting slopes are called ew-gradient (for east-west) and ns-gradient (for north-south), then the overall gradient can be calculated in the second step by: gradient = square-root( ( arctan( ew-gradient ) )2 + ( arctan( ns-gradient ) )2 ) and the overall aspect is given by aspect = arctan( ( ew-gradient ) / ( ns-gradient ) ) |
Nested Functions | A conditional statement works in much the same way as an if-then-else statement and can be used to nest functions. Output = Con (Test statement, if true do something, if false do something) For example, to identify roads that are in areas with low avalanche risk, the following statement will create a new output that only contains roads that are in low risk areas. SuitableRoads = Con (AvalancheRisk < 1, Roads) |
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.
Ready? Take the Lesson 7 Surface Analysis quiz to check your knowledge! Return now to the Lesson 7 folder in Canvas to access it. You have an unlimited number of attempts and must score 90% or more.
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.
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.
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:
For Project 7, the minimum items you are required to have in your write-up are:
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.
Analysis - Environments...should be set appropriately before doing any analysis (Figure 7.5).
In particular, use the centreCountyCivilDivisions layer as the Mask and also for the Processing Extent. You should also carefully consider what is an appropriate Cell Size for the analysis and set that parameter.
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.
Running this Euclidean Allocation (straight line distance) analysis will produce an allocation layer (Figure 7.7).
You can also request a distance layer (Figure 7.8).
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.
In your Project 7 write-up, describe how the distance analysis operation works. In your description, comment on how you would combine multiple distance analysis results (one for each high school) to produce an allocation analysis output. [Hint: First run the Euclidean Distance Tool for each high school individually and then look for a tool that when used can generate the same map that the Euclidean Allocation analysis provides (i.e., the map shown in Figure 7.7). There is a tool which could help with this -- take a look at the tools available in Spatial Analyst - Local (e.g., Cell statistics or Lowest).]
As part of your discussion, include the output maps generated by the Euclidean Allocation and Euclidean Distance processes.
Finally, comment on the differences in the Euclidean (straight-line) distance allocation and the actual allocation of places to school districts. (You will need to look at the roads, topography, and minor civil divisions to make sense of this.)
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.
There are multiple workflows you can use to produce the end result described in Step 2 above. Whichever workflow you choose, it will require multiple steps—you will have to create intermediate layers and combine those in some way to arrive at the final result as shown in Figure 7.9. The Reclassify tool in particular is your friend here, but be careful that all cells in your intermediate outputs have a value (i.e., don’t allow cells to have a value of NODATA) or you will have problematic results when you use the Raster Calculator in subsequent steps.
The end result will look something like this (although you should consider creating a better designed map):
In your Project 7 write-up, describe how you created the roads raster and the resulting output. Include any maps, figures, or tables that help you to explain your workflow.
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.)
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? Respond to these questions in your Project 7 write-up. Make sure to include the resulting map to support your explanation.
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.
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.
For your Project 7 write-up, include these estimates and explain how you went about determining the number of the school-aged children in the four school districts.
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.
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 a map showing the new 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).
Please put your write-up, or a link to your write-up, in the Project 7 drop-box.
For Project 7, the items you are required to have in your write-up are:
I suggest that you review the Lesson 7 Overview page [4] 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 [5].
Please use the Discussion - General Questions and Technical Help discussion forum to ask any questions now or at any point during this project.
You have reached the end of Lesson 7! Double-check the to-do list on the Lesson 7 Overview page [4] to make sure you have completed all of the activities listed there before you begin Lesson 8.
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 modelling 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:
LaPoint, S., Gallery, P., Wikelski, M., & Kays, R. (2013). Animal behavior, cost-based corridor models, and real corridors [6]. Landscape Ecology, 28(8), 1615-1630.
Belote, R.T., Dietz, M.S., McRae, B.H., Theobald, D.M., McClure, M.L., Irwin, G.H., McKinley, P.S., Gage, J.A. and Aplet, G.H., 2016. Identifying corridors among large protected areas in the United States [7]. PLoS One, 11(4), p.e0154223.
Links
[1] https://creativecommons.org/licenses/by-nc-sa/4.0/
[2] https://www.e-education.psu.edu/geog586/node/4
[3] https://www.nature.com/articles/srep01300
[4] https://www.e-education.psu.edu/geog586/node/646
[5] https://www.e-education.psu.edu/geog586/node/828
[6] https://link.springer.com/article/10.1007/s10980-013-9910-0
[7] https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0154223