USGS visual identity mark and link to main Web site at http://www.usgs.gov/

Digital Mapping Techniques '02 -- Workshop Proceedings
U.S. Geological Survey Open-File Report 02-370

Visualizing the Uncertainty of Geologic Maps

By Patrick J. Kennelly

Montana Bureau of Mines and Geology
Montana Tech of the University of Montana
Butte, MT 59701-8997
Telephone: (406) 496-2986
Fax: (406) 496-4451
e-mail: pkennelly@mtech.edu

ABSTRACT

For this study I quantified and displayed the uncertainty of geologic contacts located from U.S. Geological Survey (USGS) topographic base maps. I used Level 1 Digital Elevation Models (DEM's), which have a root mean square error (RMSE) of 7 meters or less. Assuming a normal distribution of error, a contact traced primarily from topographic contours has a 68.3% chance of being within ±7 vertical meters of the contact's actual location, a 95.5% chance of being within ±14 vertical meters, and a 99.7% chance of being within ±21 vertical meters. For this study I used the top and base of the Tullock Member of the Fort Union Formation in the eastern half of the U.S. Geological Survey Broadus 30 x 60 minute quadrangle in southeastern Montana as an example. The geologic contact was buffered in the z direction using the underlying digital elevation model data. Resulting areas vary in horizontal width and are represented by dark gray (1 x RMSE), medium gray (2 x RMSE), and light gray (3 x RMSE). This method could serve as a reconnaissance tool, assisting field geologists in better locating geologic contacts.

INTRODUCTION

Locational error is present in and impossible to eliminate from geologic maps, but can be visualized with geographic information system (GIS) technology. The probability of a geologic contact being precisely located is a function of such diverse factors as how easily the contact can be recognized in the field, the detail of the field work, the accuracy of the manual or digital data capture, and errors inherited from the topographic base map. Although some of these uncertainties are difficult to quantify, the horizontal and vertical errors inherent in topographic base maps available from the U.S. Geological Survey (USGS) are well documented (USGS, 1998, 2000).

UNCERTAINTY OF TOPOGRAPHIC MAPS

The United States National Map Accuracy Standards define horizontal and vertical accuracy by the associated error. To meet the horizontal accuracy standard, 90% or more of well defined test points must not be in error by more than 1/30 inch for maps printed at scales less detailed than 1:20,000. Well-defined test points include benchmarks or perpendicular road intersections, but not geologic contacts. To meet the vertical accuracy standard, 90% or more of tested elevations must fall within one-half of the contour interval. The vertical standard is appropriate for geologic contacts created from contour patterns augmented with some known points.

To measure topographic uncertainty, the USGS generally uses 28 test points to determine the vertical accuracy of a 7.5-minute sheet. From these test points, a root mean square error (RMSE) can be calculated using the following equation:

Equation

where di = Zestimated -Zobserved and n is the number of sample points. Given this RMSE, the USGS has assigned all DEM's to one of three categories. Level 1 DEM's are derived using photogrammetric techniques, and contain the least vertical error. They have a RMSE of 7 meters or less (Slocum, 1999).

The RMSE is a summary statistic for a map. It gives no indication of how error is distributed, statistically or spatially. Error curves of very different shapes (e.g., normal, skewed) could have the same RMSE. Additionally, one segment of a map could account for the majority of the error in the map (Shortridge, 2001). In this study, I made the simplifying assumptions that (1) the error has a normal or Gaussian distribution (Figure 1), and (2) this error is distributed across the map.

Normal or Gaussian distribution curve       Figure 1. A normal or Gaussian distribution curve. Values within one standard deviation account for 68.3% of the values and are shown in dark gray. Values within two standard deviations account for 95.5% of the values and include the medium-gray areas. Values within three standard deviations account for 99.7% of the values and include the light-gray areas.

LOCATION AND GEOLOGY OF THE STUDY AREA

For this study I looked at the geologic contacts associated with the top and base of the Tullock Member of the Tertiary Fort Union Formation in the eastern half of the USGS Broadus 30 x 60 minute quadrangle (scale 1:100,000) in southeastern Montana (Figure 2). The Tullock is a planar-bedded light-yellow to light-brown sandstone interbedded with shale and mudstone. Its basal contact is the base of the lowest persistent coal bed. It is underlain by a cross-bedded yellowish-gray sandstone, the Cretaceous Hell Creek Formation. The Tullock is overlain by the Lebo Member of the Fort Union Formation, a gray, smectitic shale and mudstone containing lenses of gray and yellow sandstone (Figure 3). See Vuke and others (2001) for more detailed descriptions of these geologic units.

Location of the study area       Figure 2. Location of the study area within the eastern half of the USGS Broadus, Montana, 30 x 60 minute quadrangle, southeastern Montana. The area is approximately 40 km by 55 km.

Geologic structure in this area is minimal and is characterized by low-angle bedding dipping to the northwest. The geologic contacts were constructed from a combination of field observations, regional correlations, and integration of previous mapping. In many cases contacts were interpolated between observed locations by the mapper, using knowledge of the structural orientation of the units and contours from the USGS topographic base maps.

Geologic units and topography of the eastern half of the USGS Broadus 30 x 60 minute quadrangle
 
Figure 3. Geologic units of the eastern half of the USGS Broadus, Montana, 30 x 60 minute quadrangle.       Figure 4. Topography of the eastern half of the USGS Broadus 30 x 60 minute quadrangle, overlain with contacts of the geologic units from Figure 3.

The topography in the study area ranges from elevations of 900 to 1,350 meters (Figure 4). Slopes are as steep as 37°. The Little Powder River runs through the center of the study area, and associated Quaternary alluvium overlies the bedrock geology.

METHODOLOGY

Geologic contacts were converted to line themes and overlain on the 16 Level 1 DEM's (scale 1:24,000) that compose the eastern half of the Broadus quadrangle. I applied a vertical buffer using an ArcView extension developed by Damon Holzer (Department of Forest Service, Texas A & M University) and available on ESRI's Web page http://arcscripts.esri.com/. The vertical buffer identified all grid cells within a given elevation of the contacts. Three vertical buffers were created to represent the first, second, and third standard deviations of error in the DEM's, at ±7, 14, and 21 vertical meters, respectively. Due to variations in slope, vertical buffers of equal interval define areas of variable horizontal width.

A horizontal distance of 200, 400, and 600 meters was defined as the maximum horizontal width associated with the vertical buffers of ±7, 14, and 21 meters, respectively. This definition prevented inclusion of areas that are unrealistic distances from interpreted contacts based on the geologist's assesment. The gentle dip of geologic formations was ignored in this analysis. At the maximum horizontal distance of 600 meters from the contact, dips of 5° offset horizontal locations by approximately 50 meters.

RESULTS

The resulting buffer is displayed on the geologic map in Figure 5 and the topographic map in Figure 6. Dark-gray pixels are those within 7 vertical meters; medium-gray pixels are between 7 and 14 vertical meters; and light-gray pixels are between 14 and 21 vertical meters. These shades of gray can be related to the probability, assuming a normal distribution (Figure 1). The probability of the location of the contact being within the dark-gray area is 68.3%, within the red and medium-gray area is 95.5%, and within the dark-gray, medium-gray and light-gray area is 99.7%.

Probability of the geologic contact being within the dark-gray, medium-gray, and light-gray areas displayed on the geologic map       Figure 5. Probability of the geologic contact being within the dark-gray, medium-gray, and light-gray areas displayed on the geologic map; compare with Figure 1. The areas were defined using a vertical buffer.

Probability of the geologic contact being within the dark-gray, medium-gray, and light-gray areas displayed on the topographic map       Figure 6. Probability of the geologic contact being within the dark-gray, medium-gray, and light-gray areas displayed on the topographic map; compare with Figure 1. The areas were defined using a vertical buffer.

The horizontal width of the resulting buffer is highly variable. For the third standard deviation (dark gray, medium gray, and light gray), the width ranges from 90 to 1,200 meters (using the arbitrary maximum horizontal distance of 600 meters on each side of the contact). The width is greater in gently sloping areas and lesser in steeply sloping areas. The effect of slope on the probability of properly locating the contact in cross-sectional and map views is illustrated in Figure 7, and shown in perspective view for a part of the geologic map in Figure 8.

Normal distribution of error is shown on a cross-sectional view and map view of a sample location       Figure 7. Normal distribution of error is shown on a cross-sectional view and map view of a sample location. Vertical widths are constant, but horizontal widths vary with the slope of the topography.

Figure 8. Perspective view of the east-central part of the geologic map from Figure 3. The Tullock Member overlies the Hell Creek Formation. Each frame adds a gray buffer to represent an additional standard deviation of locational error.       Perspective view of the east-central part of the geologic map from Figure 3

DISCUSSION

The method of local, vertical buffering seems most appropriate for geologic maps because it allows interpreted contacts to be used to their fullest extent. It is similar to a method devised by Hunter and Goodchild (1995) to determine the probability of an area being above a threshold elevation. Their study, however, used only one elevation value; this study allowed the elevation of the contact to vary. Several alternative methods could be used to create a vertical buffer for geologic contacts. If the geologic formations were horizontal, the buffers could be displayed directly from the DEM. This is not the case in this study area, where elevations of the contact range from 950 to 1,250 meters.

Alternatively, elevation values could be assigned to the top and basal contacts, and then a surface could be fit to the lines in three dimensions. A best-fit planar analysis in three dimensions would be equivalent to a linear regression of point data in two dimensions; then the surface could be offset by the RMSE z-values and intersected with the DEM. This method has two disadvantages. First, all segments of contact lines need not fall directly on the surfaces. Thus, there could be local variations not interpreted by the geologist. Second, new areas could be added in regions where the geologist has interpreted the contact as not present. In fact, the latter is the case in the southeastern corner of the study area. The Tertiary Tullock Member was determined to be absent on the basis of pollen samples, although the known topography and geologic structure would indicate that it should be present. The geologic interpretation is that structural dip increases in the southeastern part of the study area.

Areas inappropriate for this type of analysis include those where bedrock geologic contacts are overlain by alluvium. At these locales, the geologist has interpreted the location of the contact beneath the alluvium. Thus, the surface elevation, which now represents the top of the alluvium, is no longer appropriate. In areas where the contact is overlain by alluvium, such as in the stream valley in the northwestern part of the map, an evenly spaced concentric buffer pattern is apparent in Figure 5. The assumption that errors in elevation follow a normal curve is critical to the validity of this method. If error is random, the distribution should be normal (Wise, 1998). Systematic error, however, would not have a normal distribution. One such sampling error in the construction of DEM's is referred to as the "Firth Effect" and results in north-south or east-west lineations. This effect is caused by operators of photogrammetric equipment sampling row by row or column by column in alternating directions, and consistently underestimating elevation when moving upslope and overestimating elevation when moving downslope (Hunter and Goodchild, 1995). Visual inspection of hillshaded DEM's clearly reveal this striped pattern. These DEM's are generally not Level 1 and should be avoided for analysis.

CONCLUSIONS

The uncertainty inherited by geologic contacts from USGS topographic base maps can be quantified and visualized. The method described in this study assumes a normal or Gaussian distribution of measured error and applies vertical buffers to the contacts based on these measures. These vertical buffers represent one, two, and three standard deviations based on a measured RMSE. The map width of these buffers is variable.

This GIS method could be a reconnaissance tool used by geologists to determine the areas requiring more detailed field inspection. The method would be especially useful in larger scale mapping, such as 1:24,000 scale.

REFERENCES

Brown, J.L., 1993, Sedimentology and depositional history of the lower Paleocene Tullock Member of the Fort Union Formation, Powder River Basin, Wyoming and Montana: U.S. Geological Survey Bulletin 1917-L, 42 p.

Hunter, G.J., and Goodchild, M.F., 1995, Dealing with error in spatial databases: a simple case study: Photogrammetric Engineering and Remote Sensing, v. 61, p. 529Ð537.

Shortridge, A., 2001, Characterizing uncertainty in digital elevation models, in Hunsaker, C.T., Goodchild, M.F., Friedl, M.A., and Case, T.J., eds., Spatial uncertainty in ecology: implications for remote sensing and GIS applications: New York, Springer-Verlag, p. 238-257.

Slocum, T., 1999, Thematic cartography and visualization: Upper Saddle River, N. J: Prentice-Hall, Inc., 293 p.

U.S. Geological Survey, 2000, U.S. GeoData digital elevation models: U.S. Department of the Interior Fact Sheet 040-00: Washington, D.C., . U.S. Geological Survey, 1998, National Mapping Program technical instructions: standards for digital elevation models: U.S. Department of the Interior, .

Vuke, S.M., Heffern, E.L., Bergantino, R.N., and Colton, R.B., 2001, Geologic map of the Broadus 30- x 60-minute quadrangle, eastern Montana: Montana Bureau of Mines and Geology Open File Report #432, scale 1:100,000.

Wise, S.M., 1998, The effect of GIS interpolation errors on the use of digital elevation models in geomorphology, in Lane, S.N., Richards, K.S., and Chandler, J.H., eds., Landform monitoring, modeling and analysis: Chichester, England, John Wiley and Sons, p. 139-164.


RETURN TO Contents
National Cooperative Geologic Mapping Program | Geologic Division | Open-File Reports
U.S. Department of the Interior, U.S. Geological Survey
URL: https://pubsdata.usgs.gov/pubs/of/2002/of02-370/kennelly2.html
Maintained by David R. Soller
Last modified: 19:15:43 Wed 07 Dec 2016
Privacy statement | General disclaimer | Accessibility