The project opens in ArcGIS Pro.

The Madison Temperature map consists of the World Light Gray Canvas Base basemap and two feature layers: Temperature_Aug08_8pm and Block_Groups. The Temperature_Aug08_8pm layer contains 139 points spread across Madison, Wisconsin, covering the city center and surrounding rural areas. Each point represents the location of a sensor measuring temperature changes at 15-minute intervals. The points in the Temperature_Aug08_8pm layer represent temperature measurements in degrees Fahrenheit taken on August 8, 2016, at 8:00 p.m. at each of the sensors.

In the layer, sensor locations are symbolized in shades of yellow to red representing change in temperature in degrees Fahrenheit. The lightest shade of yellow corresponds to 73 degrees Fahrenheit (22.78 degrees Celsius), and the darkest shade of red corresponds to 86 degrees Fahrenheit (30 degrees Celsius).

The attribute table for Temperature_Aug_08_8pm appears. The table contains a record of attribute values for each of the 139 individual sensor points. The TemperatureF field maintains the temperature measurement value.

In the TemperatureF field, the highest temperature recorded is 83.869 degrees Fahrenheit and the lowest recorded value is 73.429.

The Block_Groups layer represents census block groups in the city of Madison and surrounding townships. Block groups are symbolized by the density of residents over the age of 65, calculated by dividing the population over age 65 by the area of the block group in square kilometers.

These block groups will serve as the extent of the study area for the exercise.

As a final step, you will predict the average temperature in each block group to locate areas of Madison that are characterized by both high average temperatures and a high density of residents over the age of 65.

The Chart Properties panes and a chart view appear. Initially, the chart view is empty.

The chart updates to show a histogram of temperature measurements and the chart title Distribution of TemperatureF appears. Additionally, the Statistics group in Chart Properties updates, showing various statistics for the TemperatureF histogram field.

In the chart, a blue vertical line is displayed at the mean (average) temperature value (79.4 degrees). Temperature values are spread fairly evenly between the minimum and the maximum, with the largest number of points showing a temperature between 79.5 and 81.3 degrees. The median temperature is displayed in purple and the standard deviation in brown.

In the Chart Properties pane, under Statistics, the Count value is 139 points and the Min and Max temperature values are 73.4 and 83.9 degrees, respectively.

The points with the lowest temperature measurements are selected on the Madison Temperature map. These lower temperature measurements are located mostly in the suburban and rural areas surrounding the Madison city center.

In the Madison Temperature map, most of the highest temperature measurements are located in the downtown city center area of Madison and additionally in adjacent areas to the northeast and southeast of the city center.

In the Madison Temperature map, most of the highest temperature measurements are located in the downtown city center area of Madison and additionally in adjacent areas to the northeast and southeast of the city center.

The Geostatistical Wizard appears and shows the available interpolation methods in the left pane and dataset options in the right pane.

The right side of the Geostatistical Wizard updates to show applicable Kriging/CoKriging options.

• For Source Dataset, choose Temperature_Aug_08_8pm.
• For Data Field, choose TemperatureF.

By choosing Temperature_Aug_08_8pm as the source dataset and TemperatureF as the data field, you specify that you want to perform simple kriging on the temperature measurements. By not providing a second dataset, you'll perform kriging rather than cokriging. You can learn more about cokriging in Understanding cokriging.

On the second page of the Geostatistical Wizard, you'll specify which type of kriging you want to perform and configure options applicable to that type of kriging.

This parameter specifies that you won't perform any transformations.

The Semivariogram/Covariance Modeling page appears.

This parameter updates the graph from covariance to semivariogram.

The graph on the left now updates to display a semivariogram instead of covariance. The semivariogram is the mathematical backbone of kriging and fitting a valid semivariogram is almost always the most difficult and time-consuming step in building a kriging model.

The semivariogram pane is composed of three sections:

• Semivariogram—The graph in the upper left of the pane, containing binned values (red points), averaged values (blue crosses), and the semivariogram model (blue curve).
• General Properties—The parameters in the right pane of the page, used to configure the shape of the blue semivariogram model.
• Semivariogram map—Located on the lower left of the page, used to detect anisotropy. Anisotropy will not be discussed in these tutorials.

The semivariogram is configured by three parameters that are found in General Properties:

• Nugget—The value of the semivariogram at the y-axis, which represents the expected squared difference in the value of points that are zero distance apart. While in theory the expected squared difference for these points should be zero, a nugget value greater than zero often occurs due to microscale variation and measurement errors.
• Major Range—The distance where the semivariogram becomes flat. If two points are separated by a distance larger than the major range, the points are considered uncorrelated.
• Partial Sill—The value of the semivariogram at the major range is called the sill. The partial sill is calculated by subtracting the nugget from the sill and represents the expected squared difference in value between points that are spatially uncorrelated. This value provides information about the variance of the underlying spatial process.

The goal of the semivariogram page is to configure the parameters in General Properties such that the blue semivariogram passes as closely as possible through the middle of the binned and averaged values in the semivariogram graph.

The blue semivariogram slightly changes after changing the model.

There is a lot of detail packed into the semivariogram page, and it is often difficult even for experienced geostatisticians to determine the appropriate parameters of a semivariogram. For this reason, the Optimize model button was created.

The purpose of the Optimize model button is to automate finding a nugget, major range, and partial sill that result in the smallest root mean square cross-validation error (cross-validation will be shown and explained later in this tutorial). The Geostatistical Wizard does not apply this optimization by default only because it can sometimes take a long time to calculate.

After optimizations, the semivariogram and parameters are updated. These are the values that you will use for your first kriging model.

The wizard updates to display the Searching Neighborhood page, which consists of a preview of the prediction map along with parameters that control the searching neighborhood.

Each prediction is based on neighboring input points, and this page allows you to control how many neighbors will be used and which direction the neighbors will come from. Because your temperature measurements are evenly spread over the map, the default searching neighborhood does not need to be altered. If the input points were more clustered or unevenly spaced, you would need to account for this in the searching neighborhood.

The center of the searching circle moves to the specified x,y coordinate in the middle of a hot part of the city.

Have you pinpointed the center of a heat island at this location? No. Heat islands don't really have a center—they tend to spread out across a city.

At this x,y location, Identify Result predicts that the temperature is 83.26 degrees with a standard error of 0.51 degrees. Standard errors quantify the uncertainty in the predicted values. The larger the standard error of the prediction, the higher the uncertainty in the predicted value.

• In this location, for example, the lower bound of the 95 percent confidence interval is (83.26 – 2 * 0.51) = 82.24.
• The upper bound of the confidence interval is (83.26 + 2 * 0.51) = 84.28.

Therefore, the best estimate for the temperature at this location is 83.26 degrees Fahrenheit, but you can be 95 percent confident that the true temperature is somewhere between 82.24 and 84.28 degrees Fahrenheit.

The prediction location moves to the top of the study area in the coldest part of the map. The predicted value for this location is about 75.22 degrees with a standard error of 1.76. At this location, the standard error is much larger. This is because there are fewer temperature measurements toward the top of the map than there are in the city center. This results in larger uncertainty in temperature predictions in areas with fewer measurements.

Next, you will explore the cross-validation page. The cross-validation page displays various numerical and graphical diagnostics that allow you to assess how well your interpolation model fits your data. Cross-validation is a leave-one-out validation method that sequentially hides each input point and uses all remaining points to predict back to the location of the hidden point. The measured value at the hidden point is then compared to the prediction value from cross-validation; the difference between these two values is called the cross-validation error.

The summary is useful for quickly assessing the overall accuracy and reliability of the model. Each summary statistic provides different information about the model.

Overall, these statistics are adequate to justify the accuracy of your kriging model.

• The Mean statistic indicates that on average the temperature predictions are 0.14 degrees too high, which is a small amount of bias and should not be concerning.
• The Root-Mean-Square statistic indicates that on average the predictions differed from the measured values by a little less than two degrees.
• Because the Root-Mean-Square Standardized statistic is larger than one, this indicates that the standard errors are being slightly underestimated.

The Predicted graph displays a scatterplot of the cross-validation predictions (x) versus measured values (y) for each input point. In addition, a blue regression line is fitted to the data and a gray reference line is used to compare the blue regression line to the ideal. If your interpolation model is valid, the predictions should be approximately equal to the measured values, so the regression line would follow a 45-degree angle.

In your graph, the blue regression line follows the reference line very closely, which gives you further confidence in the accuracy of your model.

Notice in the Error graph, your blue regression line is decreasing. This indicates that the interpolation model is smoothing the data, meaning that large values are being underpredicted, and smaller values are being overpredicted. Some degree of smoothing occurs in almost every geostatistical model, and in this result, smoothing is not severe.

In the Normal QQ Plot graph, if the red dots fall close to the gray reference line, it indicates that the predictions follow a normal distribution. In your graph, the red points do generally fall close to the reference line, but there are some deviations, especially for the points on the upper right part of the graph. While interpreting QQ plots is not an exact science, your graph indicates that you are justified in assuming that the predictions follow a normal distribution.

The final page of the wizard is the Method Report page, which displays all the parameters and settings that were used for the interpolation.

The Geostatistical Wizard closes and a layer named Kriging, showing predicted temperature values, is added to the Contents pane of your map.

The urban heat island effect is clear just from looking at the map. The highest predicted temperatures are in the downtown area of Madison, with temperatures generally in the range of 80 to 84 degrees. Lower predicted temperatures are in the surrounding suburban and rural areas, with temperatures in the range of 73 to 78 degrees.

As you investigate higher predicted temperature locations in the middle of the city, notice the associated lower standard errors. It is safe to assume that the predicted temperatures are higher due to the urban heat island effect, and the standard errors are lower because there are more temperature measurements in the middle of the city.

The top left pane displays a preview of the interpolated surface with a searching circle centered in the middle of the data extent.

The lower right displays Identify Result.

General Properties shows parameters for the semivariograms and the searching neighborhood.

Parameters in General Properties provide control over subsets and simulations in EBK:

• Subset Size specifies the number of points in each subset.
• Overlap Factor allows you to control how much these subsets overlap each other.
• Number of Simulations controls how many semivariograms will be simulated in each subset.

The Simulated semivariograms (blue lines) and Empirical semivariogram (blue crosses) are displayed in the lower left. The median semivariogram is solid red, and the first and third quartiles are displayed as dashed red lines.

The preview surface updates to reflect the new subset size. With 139 input points, using a subset size of 50 will create approximately three subsets. This ensures that the semivariograms will be sufficiently estimated at a local level, while still maintaining enough points in each subset to reliably estimate the semivariogram parameters.

The predicted temperature at this location is about 83.39 degrees with a standard error of 0.63 degrees. In the previous tutorial, simple kriging predicted 83.26 degrees with a standard error of 0.51 degrees at this same location.

At this location (571000, 290000), the semivariograms seem to pass through the averaged values (blue crosses) fairly well, particularly at short distances. The averaged values at the largest distances tend to be on the lower end of the spectrum, but it is most critical to properly model the semivariogram at short distances, as these are the distances that will contribute most to the predicted values.

The prediction location moves to the top of the study area in the coldest part of the map. The predicted value for this location (572000, 307000) is about 74.14 degrees with a standard error of 2.28. Simple kriging predicted about 75.22 degrees with a standard error of 1.76. This time, the two predictions differ by a full degree, but this is likely due to the larger uncertainty in the predicted values at this location. This uncertainty can be seen in the larger standard errors, different than the previous x,y location.

As with simple kriging, the cross-validation page displays summary statistics on the right and graphical diagnostics on the left. In the EBK summary statistics, there are now three additional statistics that did not appear in simple kriging:

• Average CRPS—This statistic simultaneously quantifies the accuracy and stability of the model, and it should be as small as possible. Unfortunately, it has no direct interpretation, and it can only be used to compare different interpolation models.
• Inside 90 Percent Interval—The percent of cross-validation points contained in a 90 percent prediction interval. This value should be close to 90. Your value of 89.928 is nearly perfect.
• Inside 95 Percent Interval—The percent of cross-validation points contained in a 95 percent prediction interval. This value should be close to 95. Your value of 96.403 is quite close to the ideal value of 95.

The following table shows a comparison of cross-validation summary statistics from EBK and simple kriging:

• Larger Mean and Mean Standardized values in EBK indicate that it has slightly more bias than simple kriging, but overall both models have very small amounts of bias.
• The slightly lower Root-Mean-Square value indicates that on average EBK predicts slightly more accurate temperature values.

The biggest difference in the two models is that the standard errors in EBK are much more accurate.

• The larger Average Standard Error value in EBK shows that on average, EBK is estimating larger standard errors than simple kriging.
• The nearly perfect Root-Mean-Square Standardized value in EBK (recall that ideally it should be one) indicates that these standard errors are being more correctly estimated.
• The Average Standard Error value of EBK also more closely matches that Root-Mean-Square value than it does in simple kriging.

Taken together, this is strong evidence that the EBK model is more reliable than the simple kriging model.

The graph shows predicted values from cross-validation versus measured values. The blue regression line is so close to the gray reference line that you can hardly see the reference line. In simple kriging, the regression line was not as perfectly aligned with the reference line. This should give you further confidence that the EBK model is more reliable.

Like the simple kriging model before, the blue regression line is slightly decreasing, which indicates that the model has performed smoothing of the data, but this smoothing is not severe.

The red points very closely follow the gray reference line. There is still some deviation from the reference line for the largest values, but this deviation is smaller than it was in simple kriging. Based on this graph, you can safely assume that the predictions follow a normal distribution.

The Geostatistical Wizard closes, and the Empirical Bayesian Kriging geostatistical layer is added to the Contents pane. This layer has the same symbology as the Kriging layer, so they can be visually compared.

The USA NLCD Impervious Surface Time Series layer covers the entire continental United States, but your study area covers the extent of the Madison, Wisconsin, area. As a result, you'll create a subset of the source data to the extent of your study area by using the Extract By Mask geoprocessing tool.

The Geoprocessing pane appears.

• For Input raster, choose USA NLCD Impervious Surface Time Series.
• For Input raster or feature mask data, choose Block_Groups.
• For Output raster, type Impervious_Surfaces.

The output raster will be saved in the default geodatabase of the project.

In addition to extracting Impervious_Surface values within your study area, you also want to update the coordinate system to the same projection as the rest of your data and additionally resample the source data to a more suitable cell size of 100 meters. These changes will allow faster calculations later in the tutorial.

• For Output Coordinate System, choose Block_Groups.
• For Cell Size, type 100.
• For Extent, choose Block_Groups.

The output coordinate system for the output is now set the same as the Block_Groups layer, which is NAD_1983_2011_Wisconsin_TM, and the output cell size is set to resample to 100 meters.

The control updates to As Specified Below and the output minimum and maximum extent values are updated to match the minimum and maximum extent of the Block Group layer.

The layer Impervious_Surfaces is added to the map and Contents pane. You will adjust the symbology.

Your raster layer Impervious_Surfaces is a subset of the USA NLCD Impervious Surface Time Series layer and contains extracted values covering the extent of the Block_Groups layer that are resampled to 100-meter cell size in the correct projection needed for your analysis.

You no longer need the USA NLCD Impervious Surface Time Series layer so you'll remove it.

The highest percentage of impervious surfaces are in the middle of the city and along transportation corridors, and fewer impervious surfaces are located in suburban and rural areas surrounding the city, which generally have higher percentages of vegetation and open space.

There are also no impervious surface values covering the lakes. As a result, EBK Regression Prediction will not make temperature predictions across lakes. This is desirable because all your source temperature measurements were taken on the land and are thus unlikely to reliably predict temperature over the lakes. Temperature variation across water is driven by different factors than land temperatures.

• For Input point features, choose Temperature_Aug_08_8pm.
• For Input raster, choose Impervious_Surfaces.
• For Output field name, type Raster_Value.

The Raster_Value field is added to the Temperature_Aug_08_8pm table. This Raster_Value attribute represents the impervious surface value extracted from the Impervious_Surfaces raster layer for each point location.

• For X-axis number, choose TemperatureF.
• For Y-axis number, choose Raster_Value.

The chart updates to display the scatter plot and is titled Relationship between TemperatureF and Raster_Value.

The scatter plot shows a clear positive relationship between the measured temperature (TemperatureF) and the percentage of impervious surfaces (Raster_Value). In addition, the relationship appears to be roughly linear, as the trend line appears to pass through the middle of the points. The higher the percentage of impervious surfaces, the higher the temperature. This linear relationship between the variables is important because linear regression rests on this assumption.

• For Input dependent variable features, choose Temperature_Aug_08_8pm.
• For Dependent variable field, choose TemperatureF.
• For Input explanatory variable rasters, choose Impervious_Surfaces.
• For Output prediction raster, type Temperature_Prediction.

This parameter specifies that each subset will have 50 points, which matches the values used in EBK in the previous tutorial.

The control updates to As Specified Below and the output minimum and maximum extent values are updated to match the minimum and maximum extent of the Block Group layer.

Two layers, named EBKRegressionPrediction1 and Temperature_Prediction, are added to the Contents pane.

The only layer visible is EBKRegressionPrediction1.

The EBKRegressionPrediction1 layer shows the same interpolation pattern of urban heat as both simple kriging and EBK, but it clearly has a lot more precision. The contours are more refined, and the temperature values change over much shorter distances, indicating a higher degree of accuracy. No interpolation has occurred over the lakes, and as a result, we see a more realistic temperature map, which once again needs quantitative verification using cross-validation.

The Cross validation window is identical to the final page of the Geostatistical Wizard and allows the exploration of the geostatistical layers results. Summary statistics are organized on the right and graphical diagnostics on the left.

The following table compares summary statistics for this EBK Regression Prediction as well as for the EBK and simple kriging you completed in previous tutorials:

• For EBK Regression Prediction, the Average CRPS value is about 20 percent lower than EBK, and the Root-Mean-Square value is about 25 percent lower than EBK. These are both strong indications that EBK Regression Prediction is more accurate than EBK or simple kriging.
• The smaller Mean and Mean Standardized values also show that EBK Regression Prediction has the lowest level of bias, and the Average Standard Error value is closely aligned with the Root-Mean-Square value.
• There is some evidence that the standard errors are being slightly overestimated because the Root-Mean-Square Standardized value is less than one, and the Inside 90 Percent Prediction Intervals and Inside 95 Percent Prediction Intervals contain a slightly different percentage of points than they are expected to (91.971 and 93.431 percent, respectively), but the standard errors look accurate overall.

Based on these statistics, EBK Regression Prediction is clearly the most accurate and reliable of the three kriging models.

In the Predicted graph, the regression line (blue) is almost perfectly aligned with the reference line (gray). There is a lot of variability in the points around the regression line, but this graph should give you further confidence in the accuracy of the model.

Like the two models before, the regression line in the Error graph is trending down. This indicates some smoothing in the model, but once again, the smoothing is not severe.

Points in the Normal QQ Plot graph fall closer to the reference line than in either of the previous two models. Even the largest values fall very close to the line. There is some minor deviation from the line for the smallest values, but you can safely assume that the predictions follow a normal distribution based on this graph.

Based on the numerical and graphical cross-validation diagnostics, you now have strong evidence that the EBK Regression Prediction model provides the most accurate predictions of the three models you have used in these tutorials. This is the model that will serve as your recommended procedure for interpolating temperature in Madison, Wisconsin.

Now that you have decided on using the EBK Regression Prediction model, you will apply attractive and meaningful symbology to the Temperature_Prediction raster.

You will now apply more meaningful symbology to Temperature_Prediction by importing a custom stretch renderer from an existing layer file.

The Temperature_Prediction layer symbology updates.

The EBKRP_Symbology.lyrx file contains predefined symbolization methods and properties suitable for the Temperature_Prediction layer.

The layer is symbolized with a stretched color scheme ranging from 73 degrees Fahrenheit in the lightest shade of yellow to 86 degrees in the darkest shade of red. This color ramp matches the one that was used for temperature measurement points in the Temperature_Aug_08_8pm layer.

The urban heat effect is obvious just by viewing the layer. The hottest temperatures are in the middle of the city, and the coldest temperatures are in the surrounding rural areas. However, by including the impervious surfaces layer, you are getting far greater detail in the predicted surface. In some areas, you can even pick out urban corridors and view how the heat flows between the buildings and along the highways and freeways.

• For Input raster or feature zone data, choose Block_Groups.
• For Zone field, choose OBJECTID.
• For Input value raster, choose Temperature_Prediction.
• For Output table, type Mean_Temperature.
• For Statistics type, choose Mean.

Choosing Mean for the statistics type indicates that you want to determine the average of all temperature predictions within a block group.

The table appears in the Contents pane, under the Standalone Tables section. It contains 269 records, one for each of the 269 block groups in the study area. In the table, the OBJECTID field identifies individual block groups and the Mean field contains the average predicted temperature within each block group.

Next, you will join the Mean_Temperature table to the block groups in order to add the Mean field values to each individual block group polygon.

• For Input Table, choose Block_Groups.
• For Input Join Field, choose OBJECTID.
• For Join Table, choose Mean_Temperature.
• For Join Table Field, choose OBJECTID.

Attribute fields from the Mean_Temperature table are now joined to block groups using the OBJECTID to identify each unique block group.

This field contains the average predicted temperature for each block group.

Next, you will symbolize the block groups by the predicted average temperature and apply symbology from an imported layer file.

• For Input Layer, choose Block_Groups.
• For Symbology Layer, browse to the location where you extracted the downloaded project and choose BG_temperature.lyrx.
• Under Symbology Fields, for Type, verify that the value is Value field.
• For Source Field, verify that the value is Mean_Temperature.MEAN.
• For Target Field, verify that the value is MEAN.

The block group symbology updates to show each block group polygon shaded by the average predicted temperature within that block group. The color range used is the same as the original Temperature_Aug_08_8pm layer. The average temperature follows the same patterns as the prediction raster: the hottest block groups are located in and around the center of the city, and the coldest block groups are in the surrounding suburban and rural areas.

Query expressions use the following syntax:

Field name + Operator + Value or Field

• For Input Rows, choose Block_Groups.
• For Selection type, choose New selection.

Expressions can include additional clauses or conditions that are connected to the original clause using a connector such as And or Or. Connectors indicate whether one or both clauses need to be true to select a feature.

This expression selects block groups with an average temperature above 81 degrees Fahrenheit and a density of residents over the age of 65 that is greater than 100,000 people per square kilometer.

The block groups that match the expressions are selected on the map.

Several block groups are selected based on your criteria. They are all located in downtown areas and areas along transportation corridors, and they represent the areas of the city where there is high potential for heat-related illnesses in the vulnerable population. In an emergency, these are the areas that should be prioritized by health care authorities.

As a final check, you will create a scatter plot of the average temperature versus the density of elderly residents to visualize the overall relationship.

The scatter plot updates to show the relationship between average temperature and density of elderly residents. The five selected block groups remain selected in the scatter plot and indicate occurrences where the average temperature is above 81 degrees Fahrenheit and the density of residents over the age of 65 is above 100,000.

There appears to be no relationship between average temperature and density of elderly residents. The trend line is very flat with a slightly negative slope, and the scatter plot does not show any marked patterns. This is good news, because it means that elderly residents over 65 do not tend to live in the hottest parts of Madison, Wisconsin.

The selected block group has a high density of elderly residents (over 700,000) and falls in the middle of the temperature range (around 80 degrees Fahrenheit).

Because this block group has such a high density of elderly residents, the temperature of the block group should be closely monitored by the emergency managers of Madison, Wisconsin. Fortunately, on August 8 at 8:00 p.m., this block group did not experience higher temperatures compared to the rest of Madison.