My last column, "A new standard for spatial data accuracy," March 2005, presented considerable discussion on the new National Standard for Spatial Data Accuracy (NSSDA). The column also provided information on the older accuracy standards that NSSDA was intended to replace, the National Map Accuracy Standard (NMAS) and the ASPRS Standards for Large-Scale Maps.

This month's discussion builds on the March content and provides a specific mathematical application of each of these standards. The example data in this case is from an actual project where a firm was required to provide an accuracy assessment as a project deliverable.

## The Project

Mapping of Falmouth, Ky., was performed for the Louisville District of the U.S. Army Corps of Engineers. The Corps required that the mapping meet the NMAS for maps prepared at a map scale of 1"=100' with topographic information shown at a 2-foot contour interval. Careful planning took place before arriving at a flying height and control scheme for the project. Photography was captured from a flight 4,200 feet above ground, yielding a nominal scale in the photography of 700 feet per inch. In all, some 16 GPS control points were used to control the 47 frames of photography for the project.The Corps required a formal accuracy assessment for this project that was accomplished by identifying 20 features geographically dispersed throughout the mapping that could be easily located on the ground. Because GPS surveys would be used to establish accurate three-dimensional positions of these features, care was also taken to select features that were suitable for GPS occupations. Dual-frequency static GPS observations were used to establish highly accurate positions of each of these points. In essence, the field positions of these points established with GPS were considered to be the "true" positions of the features, and therefore the differences between the mapping and field positions were also assumed to represent the error in the mapping of these features.

Table 1 lists the mapping and field (GPS) positions of each of these 20 checkpoints. All positions are shown in U.S. Survey feet. At first glance at the table, the mapping and field positions appear to match one another very well, and in fact the results are very strong. When working in a spreadsheet like Microsoft Excel, it is a simple task to determine the differences between the mapping and field positions once this information has been entered. Table 2 lists the differences for each of the 20 checkpoints. Additionally, the radial difference in the horizontal plane (determined from the individual x and y components using the Pythagorean theorem) is included in this table.

With this information tabulated, we can quickly determine if this spatial information meets the requirements of the three accuracy standards. Let's take a look at the mathematical application of each of the standards.

## National Map Accuracy Standard

The application of the NMAS is straightforward. For large-scale maps, 90 percent of the horizontal points tested must fall within 1/30 of an inch at map scale. The remaining 10 percent must fall within twice that limit, or 1/15 of an inch at map scale. For this test, the radial difference-not the individual x and y components of the error-forms the basis of the test. Since this project was intended to be presented at a map scale of 1"=100', this translates to 3.33 feet (100'/30) at the 90 percent limit and 6.67 feet for the remaining 10 percent of the points. Since the largest radial error for any of the 20 checkpoints tested was 1.27 feet, clearly the mapping greatly exceeded these horizontal requirements.
Similar to the horizontal requirements, the vertical accuracy is also tested against a 90 percent limit under NMAS. The vertical requirements are that 90 percent of the elevations tested must fall within one-half the contour interval, while the remaining 10 percent can be in error up to the contour interval. Since this mapping included a 2-foot contour interval, 90 percent of the points tested must fall within 1 foot of their true elevation, while the remaining 10 percent could be in error up to 2 feet. The largest elevation difference was only 0.47 feet, again clearly exceeding the vertical accuracy requirements of NMAS.

## ASPRS Accuracy Standards for Large-Scale Maps

One additional mathematical calculation is required for the application of both the ASPRS standards and the NSSDA. Both standards are based on the numerical Root Mean Square Error (RMSE), although in a somewhat different fashion. The RMSE is simply the square root of the average (or mean) of the squared differences between dataset (or in this case, map) values and check values. Table 3 lists the squares of the individual differences found in Table 2 and illustrates the determination of the RMSE for each of the columns.At the bottom of the table you will note the intermediate steps to the calculation of the RMSE. Each of the columns is first totaled; the average (or mean) for each component is then determined by dividing this total by the number of data points, which in this case is 20. Finally, the square root of the average of the squares is applied to determine the RMSE.

With the determination of the RMSE for each of the components, the application of the ASPRS standards becomes straightforward. To meet the requirements of these standards, the RMSE of the x and y components must be less than 1/100 of an inch at map scale while the vertical RMSE must be less than 1/3 the indicated contour interval. Note that unlike the NMAS, which looked only at the radial differences for the horizontal test, the ASPRS standards are solely concerned with the individual components of the horizontal error. Since the RMSE for the x and y components are 0.55 and 0.53 feet, respectively, and given these limits are considerably less than the 1.0 foot limit (100'/100) of these standards, it is clear that the data meets the horizontal requirements. Similarly, the RMSE of the elevations is only 0.27 feet, which is considerably better than the 0.67 foot limit (2'/3) imposed by the standards. Therefore, the accuracy of the data meets these standards.

## National Standard for Spatial Data Accuracy

As I mentioned in my March column, the NSSDA is an accuracy standard that is independent of scale and contour interval. Under the NSSDA, the accuracy expectations are either determined by the client prior to project planning or are tested and reported by the data provider at the completion of the project. This project was actually started prior to the implementation of the NSSDA. While the client's accuracy expectations for this project were defined under NMAS, it is still a worthwhile exercise to determine the level of accuracy achieved using the accuracy definitions found in the NSSDA.Accuracy under the NSSDA is defined at the 95 percent confidence interval. In other words, it is the level at which you should be confident that 95 percent of the data compiled fits within. Like the ASPRS standards, the basis of NSSDA is the RMSE. Using statistical theory, the horizontal accuracy is a multiple of 1.7308 of the radial RMSE. Similarly, the vertical accuracy is defined as a multiple of 1.9600 of the vertical RMSE. When we apply the values found in Table 3, the following project accuracies are determined:

Horizontal Accuracy

1.7308 x 0.77'

1.33 feet

Vertical Accuracy

1.9600 x 0.27'

0.53 feet

Understanding the Standards

The mathematical application of each of these standards is straightforward. The NSSDA is significantly broader than the older NMAS and ASPRS standards that were generally focused on "maps," and as such were dependent on both map scale and contour interval. Unlike these other standards, the NSSDA applies to all geospatial data regardless of how it is presented. Moreover, the NSSDA is not constrained by map scale or contour interval, which is representative of the way more projects are presented in today's digital world.

Considerable information is available online regarding the NSSDA, NMAS and ASPRS standards. Both the Federal Geographic Data Committee site at www.fgdc.gov and the Federal Emergency Management Agency site at www.fema.gov are great resources. Use them to your advantage.