MaNIS/HerpNet/ORNIS Georeferencing Guidelines 4

 Table of Contents Determining Latitude & Longitude Determining Error Distance from Uncertainties Summary Appendices Glossary Essential Attributes of Coordinate Data Calculation Examples References
This document contains information about assigning geographic coordinates, and maximum error distances for those coordinates, to locality descriptions. This document does not attempt to describe the tools and methods for finding named places on maps or in gazetteers. The process of assigning coordinates and errors, called "georeferencing" or "geocoding", can be rather complicated. The complexity of the process can be greatly reduced and the consistency of the results can be greatly increased by establishing some guidelines that cover most commonly encountered locality descriptions. The guidelines for assigning geographic coordinates are presented, with examples, in the first section, Determining Latitude & Longitude.

There are several fundamental sources of uncertainty in locality descriptions. These uncertainties vary in magnitude as well as in their interactions with each other. It is essential during georeferencing to determine and record the net uncertainty of a geographic coordinate determination. This document will explain the methodology for expressing the uncertainty as a single measurement - a maximum error. There are numerous ways in which this maximum error might be expressed, but the most convenient is as a simple distance. The sources of uncertainty, their relative magnitudes, and the interactions between them are discussed in the second section, Determining Maximum Error Distance from Uncertainties.

An Appendix containing a glossary, references, and a description of the data that should be captured for each georeferenced locality is appended for the convenience of the reader.

Determining Latitude & Longitude
Geographic coordinates can be expressed in a number of different coordinate systems (e.g. decimal degrees, degrees minutes seconds, degrees decimal minutes, UTM, etc.). Conversions can be made readily between coordinate systems, but decimal degrees provide the most convenient coordinates to use for georeferencing for no more profound a reason than a locality can be described with only two attributes - decimal latitude and decimal longitude.

Named Places
The simplest locality descriptions consist of only a named place. Many gazetteers record the location of the main post office of a populated place, unless that place is a county seat, in which case the location of the courthouse is recorded. Use the same method when making measurements on a map from a populated place. Use the geographic center for the latitude and longitude of other named places. Some gazetteers give bounding boxes to describe the extents of large places. Use the distance from the coordinates of the named place to the furthest point within the named place as the maximum error distance. If the geographic center of the named place is not within the confines of the shape of the named place, use the point nearest to the geographic center that lies within the shape. In each of these cases it is best to record the method of determination of the coordinates and error in the remarks for the determination (e.g., "measured from the main post office" or "measured from the geographic center of Bakersfield").
Example: "Bakersfield

Township Range Section (TRS) descriptions are essentially no different from that of any other named place. It is necessary to understand TRS descriptions and how they describe a place. See the References section, below, for links to TRS information.
Example: "E of Bakersfield, T29S R29E Sec. 34 NE 1/4

Offsets
Offsets generally consist of combinations of distances and directions from a named place. Use the geographic coordinates of the named place (see the Named Places section, above) as a starting point. Sometimes the locality description gives a method for determining the offset (e.g., "by road", "by river", "by air", "up the valley", etc.). No special remark about how the coordinates and error were determined is necessary in cases where the measurement method is given in the locality description.
Example: "10 mi E (by air) Bakersfield

Localities that have two orthogonal measurements in them should be treated as if the measurements are "by air."
Example: "2 mi E and 1.5 mi N of Bakersfield

If the distance was along a linear feature such as a road or river, measure along the feature rather than use a straight line. There is no uncertainty due to direction imprecision (see the Precision section, below) for this type of locality description.
Example: "13 mi E (by road) Bakersfield

There is a long-standing convention that the left and right sides of a river are from the perspective of facing downstream. (Note: The text incorrectly stated that the convention was based on the perspective facing the source and was amended on 17 Jan 2004. Thanks to Margaret Thayer of FMNH for this correction.)  Thus, in the following example, the locality is on the east side of the river, in Illinois, rather than on the west side, in Missouri.
Example: "left bank of the Mississippi River, 16 mi downstream from St. Louis

Localities that have one linear offset measurement from a named place, but that do not specify how that measurement was taken are open for case-by-case judgment.
Example: "10.2 mi E of Yuma

The judgment itself must be documented in the remarks for the determination (e.g., "Assumed 'by air' - no roads E out of Yuma", or "Assumed 'by road' on Hwy. 80"). If there is no clear best choice, use the midpoint between the two possibilities as the geographic coordinate and assign an error large enough to encompass the coordinates and uncertainties of both methods. In this case, the remark should be something like "Error encompasses both distance by air and distance by road (Hwy. 80)".

Vagueness
At times, locality descriptions are fraught with vagueness. It is not the purpose here to belittle localities of this type; in fact, an honest admission of the unknown is preferable to masking it with unwarranted precision.
The most important type of vagueness in a locality description is one in which the locality is in question. No such locality should be georeferenced.
Example: "Bakersfield?

Many locality descriptions imply an offset from a named place without definitive directions or distances. In these cases use the geographic center of the named place for the geographic coordinates. For the maximum error distance, use the greatest distance that is not likely to be considered in the area of another named place. Clearly there is a measure of subjectivity involved here. Let common sense prevail and document the assumptions made.
Example: "near Bakersfield

Sometimes offset information is vague either in its direction or in its distance. If the direction information is vague, record the geographic coordinates of the center of the named place and include the offset distance in the determination of the maximum error distance.
Example: "5 mi from Bakersfield

Uncertainty in the offset distance is a fact of the business; most localities are recorded without
error estimates. The addition of an adverbial modifier to the distance part of a locality description, while an honest observation, should not affect the determination of the geographic coordinates or the maximum error. For the example below, treat the locality as if it read "3 mi E of Bakersfield."
Example: "about 3 mi E of Bakersfield

The worst type of locality description to georeference is one that is internally inconsistent. There are numerous possible causes for inconsistencies. Rather than determine coordinates for such localities, annotate the locality with the nature of the inconsistency and refer the locality to the source institution for reconciliation. One common source of inconsistency in locality descriptions comes from trying to match elevation information with the rest of the description.
Example: "10 mi W of Bakersfield, 6000 ft" (There is no place anywhere near 10 mi W of Bakersfield at an elevation of 6000 ft).

Another common source of inconsistency occurs when the locality description does not match the geopolitical subdivision of which it is supposed to be a part. At times the locality can still be determined because the geopolitical subdivision is clearly at fault. In this case, georeference the locality and annotate it to describe the problem.
Example: "Delano, Tulare Co." (Delano is in Kern Co.)

Often there is no way to know if the geopolitical subdivision or something in the locality description itself is at fault. In the following example the county may be wrong, the distance may be wrong, or the direction may be wrong. This locality cannot be disambiguated without further research, which is best left to the institution that provided the locality description.
Example: "5 mi N of Delano, Kern Co." (5 mi N would put the locality in Tulare Co.)

Determining Maximum Error Distance from Uncertainties
The three basic sources of coordinate information used in georeferencing are maps, gazetteers, and localities already recorded with coordinates. The process of georeferencing includes an assessment of the uncertainties in geographic coordinate determinations. Uncertainties may arise due to combinations of the following sources:

1) the extent of a locality
2) GPS accuracy
3) unknown datum
4) imprecision in distance measurements
5) imprecision in coordinate measurements
6) map scale
7) imprecision in direction measurements

It is essential to understand how each of these sources contribute to the net error in a geographic coordinate determination. The first six of these are sources of distance uncertainty. Though distance uncertainties may generally be summed, there are exceptions, and the georeferencer must be cognizant of them. Distance uncertainties must never be summed with direction uncertainties; the interactions between these two different types of sources are always non-linear.  The types of uncertainty that may apply for each of the sources of coordinate information are as follows:

map - locality extent, unknown datum, distance imprecision, map scale, and coordinate imprecision, which must all be summed before calculating their interaction with direction imprecision.

gazetteer - locality extent, unknown datum, distance imprecision, and coordinate imprecision, which must be summed before calculating their interaction with direction imprecision.

coordinates - unknown datum and coordinate imprecision, which must be summed to get the maximum error distance.

GPS - unknown datum, GPS accuracy, and coordinate imprecision, which must be summed to get the maximum error distance.

Guidelines for determining the magnitude of each of the types of uncertainty are given in the paragraphs below. Following the guidelines for the individual types of uncertainty are two sections on the combinations of uncertainties.

Uncertainty due to the extent of a locality
Named places are not single points; they have extents. Although there are conventions for placing the coordinates of a named place at the post office, courthouse, or geographic center of a town, one cannot be sure that the person who recorded the locality used a particular convention. Use the distance from the geographic center of the named place to its furthest extent as the uncertainty. Since many localities are based on cities that have changed in size over the years, the extents on current maps might not reflect the extents when specimens were collected. In almost every case the current extents will be greater than the historical ones and the error calculations will be overestimated. This is consistent with our conservative approach to georeferencing.
Townships are just another instance of a named place (see the References section, below, for links to information on townships). In general, a township is a 6 mile square, with each of its component sections being a 1 mile square. Exceptions to this rule occur often in practice to account for the curvature of the earth, but it is generally safe to use the values in the table below as the extents of various divisions of a township. These values assume that the greatest extent is from the center of the division to a corner of that division, and that the coordinates of the division are not being calculated from orthogonal offsets.
Note that the tools for calculating coordinates for TRS locations only calculate based on a Township Section. If you want the center of a Township or another location in a Township more precisely defined than a Section, you'll need to make calculations based on orthogonal offsets from the center of a Township Section. For example, to find the center of a Township, first find the coordinates of the center of Section 15 in that Township. Then calculate the coordinates of a point 0.5 mi W and 0.5 mi S of the center of Section 15. This new point is the center of the Township. To calculate the maximum error distance for this example, make sure to use a distance precision of "exact" and an extent of 3 mi. This will insure that the contribution to the error based on the extent of the Township will be 4.243 mi (see the Extents of divisions of a township Table, below).
To find the center of a quarter section, first find the center of the Section. Then calculate the coordinates of the quarter section by using offsets of 0.25 mi in the appropriate directions from these coordinates. For example, the center of the NW 1/4 of Section 13 would be 0.25 mi N and 0.25 mi W of the center of Section 13. In calculating the maximum error distance for this example, set the distance precision to "exact" and set the extent to 0.25 mi. This will insure that the contribution to the error based on the extent of the Township wil be 0.354 mi (see the Extents of divisions of a township Table, below). This same method can be extended to calculate the centers of other divisions of a Section as well. For 1/4 of 1/4 of a Section, use an extent of 0.125 mi, and for 1/4 of 1/4 of 1/4 of a Section, use 0.625 mi for the extent.

 Division Example Extent Orthogonal Extent Township T6S R14E 4.243 mi 3 mi Section T6S R14E Sec. 23 0.707 mi 0.5 mi 1/4 Section T6N R14E Sec. 23 NE 1/4 0.354 mi 0.25 mi 1/4 of 1/4 Section T6N R14E Sec. 23 NE 1/4 SW 1/4 0.177 mi 0.125 mi 1/4 of 1/4 of 1/4 Section T6N R14E Sec. 23 NW 1/4 NE 1/4 SW 1/4 0.089 mi 0.0625 mi

Uncertainty due to GPS accuracy
The accuracy of the coordinate data reported by a GPS varies with time, place, and equipment used. Previous to the order to cease Selective Availability (deliberate GPS signal scrambling) at 8PM EST 1 May 2000, uncorrected GPS receivers were subject to artificial inaccuracies of about 100 meters. Today, many GPS receivers have a function to determine the estimated accuracy of given reading, but this information is not universally available, nor is it often recorded with the coordinates. It is not possible to determine the actual accuracy of a GPS reading retroactively if it was not recorded at the time of the reading. In fact, many GPS receivers estimate accuracy poorly. My Garmin eTrex Summit, for example, reports positions with putative accuracies of 7 meters that are demonstrably off by 15 meters. Where extreme accuracy is required, be sure of the capabilities of your GPS under the prevailing conditions when the coordinates are recorded. For retrospective uncertainty estimates where detailed information is not available, 30 meters is a reasonable, conservative estimate of GPS accuracy in the absence of Selective Availability.

Uncertainty due to an unknown datum
Seldom in natural history collections have geographic coordinates been recorded along with the underlying datum on which they were determined. Even now, when GPS coordinates are being recorded seemingly as definitive locations, the geodetic datum is being ignored. Interestingly, even the GNIS placename data are derived from maps in both NAD27 and NAD83 without direct reference to which datum was used for any particular record. Without recording the datum with the coordinates, uncertainty is being introduced. Figure 1 shows the magnitude of uncertainty (in meters) over North America based on not knowing the datum on which the source was based.

Figure 1. Map of North America showing the magnitude of uncertainty from not knowing whether coordinates were taken from a source using NAD27, NAD83, or WGS84.

The uncertainty due to an unknown datum is a complicated, non-linear function of latitude and longitude. This map can be used as a rough guide for determining the magnitude of that uncertainty in North America. If the locality is outside the area covered by the map in Figure 1, use 1 km as the uncertainty due to an unknown datum.

Precision
Precision can be difficult to gauge from locality descriptions; it is seldom, if ever, explicitly recorded. Furthermore, a database record may not reflect, or may reflect incorrectly, the precision inherent in the original measurement, especially if the locality description has undergone interpretation from the original, verbatim description. Different precision issues arise from recording distance measurements, directions (headings), and coordinates. Potential uncertainties from each of these sources are discussed in the paragraphs below.

Uncertainty associated with distance precision
Distance may be recorded in a locality description with or without significant digits, and those digits may or may not be warranted. A conservative way to insure that distance precision is not inflated is to treat distance measurements as integers with fractional remainders. Thus 10.25 becomes 10 1/4, 10.5 becomes 10 1/2, etc. Calculate the uncertainty for these distances based on the fractional part of the distance, using 1 divided by the denominator of the fraction.
Example: "10.5 mi N of Bakersfield" (the fraction is 1/2, uncertainty should be 0.5 mi)
Example: "10.6 mi N of Bakersfield" (the fraction is 6/10, uncertainty should be 0.1 mi)
Example: "10.75 mi N of Bakersfield" (the fraction is 3/4, uncertainty should be 0.25 mi)

For distances that appear as integer powers of ten (10, 20, 300, 4000), use 0.5 times ten to that power for the uncertainty.
Example: "10 mi N of Bakersfield" (the uncertainty should be 5 mi)
Example: "140 mi N of Bakersfield" (the uncertainty should be 5 mi)
Example: "100 mi N of Bakersfield" (the uncertainty should be 50 mi)
Example: "2000 m N of Bakersfield" (the uncertainty should be 500 m)

Note: If the locality description contains distance measures of varying precision, the person recording the data was demonstrably sensitive to the highest level of precision expressed in the locality, and it is fair to assume that level of precision would not vary within a single locality description. Therefore, the distance precision for the locality can safely be assumed to be the same as for the highest precision measurement in the locality description.

Uncertainty associated with directional precision
Direction is almost always expressed in locality descriptions using cardinal or intercardinal directions rather than degree headings. This traditional practice can introduce unfortunate directional imprecisions. The problem arises from the fact that we don't know, out of context, what the recorder meant by, say, "north", except that it is probably distinct from the other cardinal directions. Hence, "north" is not "east" or "west", but it could be any direction between northeast and northwest. The directional uncertaintyin this case is 45 degrees in either direction from the given direction.
Example: "10 mi N of Bakersfield

Of course, if the locality description contains two orthogonal directions, then the measurements are likely to have been made on a map in exactly those directions. In this case, directional imprecision can be ignored.
Example: "10 mi N and 5 mi E of Bakersfield

If the locality description contains directions more specific than the cardinal directions (e.g., "NE"), then the person recording the data was demonstrably sensitive to intercardinal directions and the directional precision can safely be assumed to be greater than if only cardinal directions were used. Thus, "NE" could mean any direction between ENE and NNE, which is twice as precise as the first direction example, above. The directional uncertainty in this case is 22.5 degrees in either direction from the given direction.
Example: "10 mi NE of Bakersfield

A locality description that contains further refined directions is correspondingly more precise. Thus, in the following example the directional uncertainty is 11.25 degrees.
Example: "10 mi ENE of Bakersfield

Uncertainty associated with coordinate precision
When recording geographic coordinates, always keep as many digits as possible. Recording coordinates with insufficient precision can result in unnecessary uncertainties. The magnitude of the uncertainty is a function not only of the precision with which the data are recorded, but also a function of the datum and the coordinates themselves. This is a direct result of the fact that a degree does not correspond to the same distance everywhere on the surface of the earth. Uncertainty due to the precision with which the original coordinates were recorded can be estimated as follows:

uncertainty=sqrt( lat_error2 + long_error2)

where

lat_error = pi*R*lat_long_precision/180.0
long_error = pi*X*lat_long_precision/180.0

R is the radius of curvature of the meridian at the given latitude,
X is the distance from the point to the polar axis orthogonal to the polar axis, and
lat_long_precision is the precision with which the coordinates were recorded, as a fraction of one degree.

R is given by the following equation:

R = a (1-e2)/(1-e2sin2(latitude))3/2

where
a is the semi-major axis of the reference ellipsoid (the radius at the equator),
e is the first eccentricity of the reference ellipsoid, defined by the following equation:

e2 = 2f - f2

where
f is the flattening of the reference ellipsoid.

X is also a function of geodetic latitude and is given by the following equation:

X = abs(Ncos(latitude))

where
N is the radius of curvature in the prime vertical at the given latitude and is defined as follows:

N = a/sqrt(1-e2sin2(latitude))

Example: Lat: 10.27   Long: -123.6
In this example the lat_long_precision is 0.01 degrees. Thus, lat_error = 1.1061 km, long_error = 1.0955 km, and the uncertainty resulting from the combination of the two is 1.5568 km. Note: The foregoing method uses a semi-major axis of 6378137.0 meters and a flattening of 1/298.25722356 based on the WGS84 datum1. Following is a table showing examples of error contributions for different levels of precision in the original coordinates using the WGS84 reference ellipsoid. Calculations are based on the same degree of imprecision in both coordinates and are given for several different latitudes.

 Precision 0 degrees Latitude 30 degrees Latitude 60 degrees Latitude 85 degrees  Latitude 1.0 degrees 156904 m 146962 m 124605 m 112109 m 0.1 degrees 15691 m 14697 m 12461 m 11211 m 0.01 degrees 1570 m 1470 m 1247 m 1122m 0.001 degrees 157 m 147 m 125 m 113 m 0.0001 degrees 16 m 15 m 13 m 12 m 0.00001 degrees 2 m 2 m 2 m 2 m 1.0 minutes 2615 m 2450 m 2077 m 1869 m 0.1 minutes 262 m 245 m 208 m 187 m 0.01 minutes 27 m 25 m 21 m 19 m 0.001 minutes 3 m 3 m 3 m 2 m 1.0 seconds 44 m 41 m 35 m 32 m 0.1 seconds 5 m 5 m 4 m 4 m 0.01 seconds 1 m 1 m 1 m 1 m

Uncertainty due to map scale
Each map has an inherent level of accuracy. For most maps, particularly old ones, accuracy of scale is not documented.  However, the USGS does have horizontal accuracy standards that are supposed to be met for all of its maps. Specifically,

"For maps on publication scales larger than 1:20,000, not more than 10 percent of the points tested shall be in error by more than 1/30 inch, measured on the publication scale; for maps on publication scales of 1:20,000 or smaller, 1/50 inch."2

It is extremely important to note that a digital map is not more accurate than the original from which it was derived, nor is it more accurate when you zoom in on it. The accuracy is strictly a function of the scale of the original map. Below is a table showing the uncertainty due to scale for USGS maps.

 Scale Uncertainty (ft) Uncertainty (m) 1:1200 3.3 ft 1.0 m 1:2400 6.7 ft 2.0 m 1:4800 13.3 ft 4.1 m 1:10,000 27.8 ft 8.5 m 1:12,000 33.3 ft 10.2 m 1:24,000 40.0 ft 12.2 m 1:25,000 41.8 ft 12.8 m 1:63,360 106 ft 32.2 m 1:100,000 167 ft 50.9 m 1:250,000 417 ft 127 m

The uncertainty inherent in non-USGS maps may not be documented. Use a value of 1mm of uncertainty on non-USGS maps. 1mm is the specified chart accuracy of all NOAA nautical charts"3, and corresponds to about three times the detectable graphical error and should serve well as an uncertainty estimate for most maps. By this rule, the uncertainty for a map of scale 1:25,000, for example, would be 25 meters and that for a 1:500,000 map would be 500 meters.

Combinations of uncertainties: distances
Distance uncertainties in any given direction are linear and additive. Following is an example of a simple locality description and an explanation of the manner in which multiple sources of uncertainty interact to result in an overall maximum error distance.
Example: "6 km E (by road) of Bakersfield

In the example above, there is no error contribution from direction; all uncertainties apply only to the distance. The possible sources of uncertainty for this example are 1) the extent of Bakersfield, 2) unknown datum, 3) distance imprecision, and 4) map scale. Suppose the center of Bakersfield is 3 km from the eastern city limit and the distance is being measured on a USGS map at 1:100,000 scale on the NAD83 datum. The uncertainty due to the extent of Bakersfield is 3 km, there is no uncertainty due to an unknown datum (assuming the datum is recorded with the coordinates), the distance imprecision is 1 km, and the uncertainty due to map scale is 51 meters. The maximum error distance for this locality is the sum of these, or 4.051 km

If more than one direction is used in the locality description, uncertainties apply to each of the two cardinal directions and the combination of them is non-linear.
Example: "6 km E and 8 km N of Bakersfield"

This example consists of two orthogonal distance measures from a named place, each with its own uncertainty due to distance imprecision. Ignore for the moment all sources of uncertainty except those arising from distance imprecision. Under this simplification, a proper description of the uncertainty is a bounding box centered on the point 6 km E and 8 km N of Bakersfield.  Each side of the box is 2 km in length (1 km uncertainty in each cardinal direction from the center). Since we are characterizing the maximum error as a single distance measurement, we need the circle that circumscribes the above-mentioned bounding box. The radius of this circle is the distance from the center of the box to any corner. The radius could either be measured on a map or calculated using a right triangle, the hypotenuse of which is the line between the center of the bounding box and a corner. Given the rule that the distance precision is the same in both cardinal directions, the triangle will always be a right isosceles triangle and the hypotenuse will always be the the square root of 2 times the distance precision. So, for the above example the error distance associated with only the distance precision would be 1.414 km.

So far we have accounted only for distance imprecision for this example. How do we take into account the uncertainty due to the shape of the named place? There are many methods that could be used to determine the coordinates and error for this situation.
Note: The method presented here is quite conservative, resulting in errors larger than they need to be. A better alternative would be to multiply only the distance precision error by the square root of 2 (contributions in both dimensions), and then sum that with all other sources, which already account for the two dimensions. This second method is the one used in the Georeferencing Calculator since version 20130205.

Determine the furthest distance within the named place from the geographic center of the named place in either of the two cardinal directions mentioned in the locality description. Add this distance to the distance precision and take the square root of 2 times the sum to get the maximum error distance associated with the combination of distance precision and the extent of the named place. For the example above, suppose the furthest extent of the city limits of Bakersfield either E or N from the geographic center is 3 km. There would be a total of 4 km of uncertainty in each of the two directions and the radius of the circumscribing circle would be 4 km times the square root of 2, or 5.657 km.

What other sources of uncertainty need to be accounted for in this example? Suppose the coordinates for Bakersfield were taken from the GNIS database, in which there is no reference to datum and the coordinates are given to the nearest second. The coordinates for Bakersfield are given in the GNIS database as 35d 22' 24" N, 119d 01' 04" W. At this location the uncertainty due to an unknown datum is 79 meters. The datum uncertainty contributes in each of the orthogonal directions. Thus, the uncertainty in each direction would be 4.079 km and the net uncertainty is this number times the square root of two, or 5.769 km.

The coordinates in the GNIS database are given to the nearest second. Based on the Uncertainty associated with coordinate precision section, above, the uncertainty due to coordinate precision alone is about 39 meters at the latitude of Bakersfield. This number already accounts for the contributions in both cardinal directions, so it must not be multiplied by the square root of 2. Instead, simply add the coordinate precision uncertainty to the calculated sum of uncertainties from the other sources. For the example above, the maximum error distance is just 5.769 + 0.039 = 5.808 km.

If the coordinates for Bakersfield had been taken from a USGS map with a scale of 1:100,000, the datum would be on the map, so there would be no contribution to the error from an unknown datum (assuming the georeferencer records the datum with the coordinates). However, the uncertainty due to the map scale would have to be considered. For a USGS map at 1:100,000 scale, the uncertainty is 167 feet, or 50 meters. In the above example, the net uncertainty in each direction would be 4.050 km. When multiplied by the square root of 2 their combination would be 5.728 km. Add the uncertainty due to coordinate imprecision to get the maximum error distance. Suppose the minutes are marked on the margin of the map and you interpolated to the nearest tenth of a minute. The coordinate precision would be 0.1 minutes and the uncertainty would be about 0.239 km from this source, therefore the maximum error distance would be 5.768 + 0.239 = 5.967 km.

Combinations of uncertainties: distance and direction
Though distance imprecisions in a given direction are linear and additive, the sum of them contributes non-linearly to the error arising from directional imprecision. An additional technique is required to account for the correlation between distance and direction uncertainties.
Example: "10 km NE of Bakersfield

If we don't consider distance imprecision, the uncertainty due to the direction imprecision is encompassed by an arc centered 10 km (d) from the center of Bakersfield (at x,y) at a heading of 45 degrees (θ), extending 22.5 degrees in either direction from that point. At this scale the distance (e) from the center of the arc to the furthest extent of the arc (at x',y') at a heading of 22.5 degrees (θ') from the center of Bakersfield can be approximated by the Pythagorean Theorem.

e=sqrt( (x'-x)2 + (y'-y)2 )

where
x = d cos(θ), y = d sin(θ), x' = d cos(θ'), and y' = d sin(θ').
For the example above the error is 3.90 km.

Now let's consider the distance uncertainties in this example. Suppose the contributions to  distance uncertainty are 3 km (extent of Bakersfield), 1 km (distance imprecision for "10 km"),  0.079 km (unknown datum, coordinates are from the GNIS database), and 0.040 km (gazetteer  data are recorded to the nearest second) for a sum of 4.119 km. The error region (the shape of the region describing the combination of distance and direction uncertainties) will be a band twice this width (2 x 4.119 = 8.238 km) centered on 10 km offset arc spanning 22.5 degrees on either side of 45 degrees.

The error distance is still
e=sqrt( (x'-x)2 + (y'-y)2 )

but now x = d cos(θ), y = d sin(θ), x' = (d+d') cos(θ'), and y' = (d+d') sin(θ')
where d' is the sum of the distance uncertainties.

The geometry can be generalized, and simplified, by rotating the image above so that the point (x',y') is on the x axis.

The error distance is still
e=sqrt( (x'-x)2 + (y'-y)2 )

but now x = d cos(α), y = d sin(α), x' = d+d', and y' = 0
where d' is still the sum of the distance uncertainties and α is an angle equal to the magnitude of the direction uncertainty. For our example above, the distance uncertainty is 4.119 km and the direction uncertainty is 22.5 degrees. Given these values, the maximum error distance is 6.210 km.

Summary
Locality descriptions are inexact and seldom give estimates of uncertainty. An ideal description of a specific locality has no uncertainty. One way to achieve this ideal is to describe the locality by a complex shape within which the whole locality must certainly lie. The capture of shape data is certainly possible with current GIS technology, and is even demonstrably more efficient than the methods described above. However, there are technical challenges yet to be met in order to make the capture of shape data feasible in a collaborative, Internet-based georeferencing environment.
One alternative is to describe a locality using a point of arbitrarily high precision with an attendant maximum error distance, which encompasses all of the uncertainties associated with the geographic coordinates. This is an expression of the locality which satisfies the requirement that the locality given in the text description must lie within the spatial description. There is a fundamental advantage of this alternative; the uncertainties are summarized as a single value, the meaning of which is independent of any geodetic surface. This allows for quantitative filtering on spatial data quality, which ultimately allows the user of the spatial data to select appropriate subsets of the data for analyses.

Appendix

Datum - A geodetic datum describes the size, shape, origin, and orientation of a coordinate system for mapping the surface of the earth. In this document we use this term to refer to the horizontal datum, not the vertical datum, which is a model upon which elevations are based.

Decimal degrees - degrees expressed as a single real number (e.g., -22.343456) rather than as a composite of degrees, minutes, seconds, and direction (e.g., 7d 54' 18.32" E).

Geodetic surface model - a geometric description of the surface of the earth.

Geodetic radius - the distance from a point on the geodetic surface to the intersection of the geodetic vertical with the geodetic equatorial plane.

Geographic coordinates - latitude and longitude, measured in any of various coordinate systems.

Geographic center - To find the geographic center of a shape, first find the extremes of both latitude and longitude within the shape and then take their respective means. If the result is not within the shape itself, choose instead the point in the shape nearest to the calculated center.

UTM - Universal Transverse Mercator. A grid coordinate system specifying a datum, zone, and offsets from the equator and from the meridian of the zone. See the References section, below, for more information.

Essential Attributes of Coordinate Data
Following are the essential attributes to be captured for each locality while georeferencing.  DecimalLatitude - the latitude coordinate (in decimal degrees) at the center of a circle encompassing the whole of a specific locality. Convention holds that decimal latitudes north of the equator are positive numbers less than or equal to 90, while those south are negative numbers greater or equal to -90.
Example: -42.5100 degrees (which is roughly the same as 42d 30' 36" S).

DecimalLongitude - the longitude coordinate (in decimal degrees) at the center of a circle encompassing the whole of a specific locality. Decimal longitudes west of the Greenwich Meridian are considered negative and must be greater than or equal to -180, while eastern longitudes are positive and less than or equal to 180.
Example: -122.4900 degrees (which is roughly the same as 122d 29' 24" W).

Uncertainty (m) - the upper limit of the distance from the given latitude and longitude describing a circle within which the whole of the described locality must lie, in meters.

GeodeticDatum - the geometric description of a geodetic surface model (e.g., NAD27, NAD83, WGS84). Datums are often recorded on maps and in gazetteers, and can be specifically set for most GPS devices so the waypoints match the chosen datum. Use "not recorded" when the datum is not known.

VerbatimCoordinateSystem - the coordinate system in which the raw data are being entered. For the purpose of collaborative georeferencing this value will be "decimal degrees." However, existing geographic coordinates may be entered in degrees minutes seconds, degrees decimal minutes, or UTM coordinates. Original coordinates should be recorded as well if they are different from decimal degrees.

GeoreferenceSources - the reference source (e.g., the specific map, gazetteer, or software) used to determine the coordinates. Such information should provide enough detail so that anyone can locate the actual reference that was used (e.g., name, edition or version, year). Map scales should be recorded in the reference as well (e.g., USGS Gosford Quad map 1:24000, 1973).

GeoreferenceProtocol - a citation of a publicly available document describing the methods of georeferencing that were followed.

GeoreferencedBy - the person or organization making the coordinate and error determination.

GeoreferenceDate - the date on which the determination was made.

GeoreferenceRemarks - comments on methods and assumptions used in determining coordinates or errors when those methods or assumptions differ from or expand upon the accepted guidelines.

Calculation Examples
Following are detailed examples of uncertainty calculations, which illustrate the concepts presented in this document, with labels updated to match the latest version of the Georeferencing Calculator Georeferencing Calculator, which is designed to make these calculations easy.

Coordinates Only
Example 1: 35 degrees 22' 24" N, 119 degrees 1' 4" W
Locality Type: Coordinates only
Coordinate Source: locality description
Coordinate Format: degrees minutes seconds
Latitude (input): 35 degrees 22' 24" N
Longitude (input): 119 degrees 1' 4" W
Datum: not recorded
Coordinate Precision: nearest second
Measurement Error: 0
Distance Units: km, m, mi, yds, or ft
Decimal Latitude: 35.3733333
Decimal Longitude: -119.0177778
Uncertainty (m): 3085 m (3.085 km, 1.91693 mi, 33373.8 yds, 10121 ft, or 1.66578 nm)
Uncertainty sources: 3045 m from unknown datum at that location, 40 m from coordinate precision nearest second

Locality Type: Coordinates only
Coordinate Source: Digital USGS map: 1:24,000
Coordinate Format: decimal degrees
Latitude (input): 35.37
Longitude (input): -119.02
Coordinate Precision: 0.01 degrees
Measurement Error: 0 (digital map - no measurement)
Distance Units: km, m, mi, yds, or ft
Decimal Latitude: 35.37
Decimal Longitude: -119.02
Uncertainty (m): 1446 m (1.446 km, 0.899 mi, 1582 yds, 4745 ft, or 0.781 nm)
Uncertainty sources: 12 m from coordinate source USGS 1:24,000 map, 1434 m from coordinate precision 0.02 degrees

Geographic feature only
Example: Bakersfield
Suppose the coordinates for Bakersfield came from the GNIS database (a gazetteer) and the distance from the center of Bakersfield to the furthest city limit is 3 km.
Locality Type: Named place only
Coordinate Source: gazetteer
Coordinate Format: degrees minutes seconds
Latitude (input): 35 degrees 22' 24" N
Longitude (input): 119 degrees 1' 4" W
Datum: not recorded
Coordinate Precision: nearest second
Measurement Error: 0
Distance Units: km
Decimal Latitude: 35.3733333
Decimal Longitude: -119.0177778
Uncertainty (m): 6085 m
Uncertainty sources: 3045 m from unknown datum, 40 m from coordinate precision nearest second, 3000 m from the extent of the named place

Distance Only
Example: 5 mi from Bakersfield
Suppose the coordinates for Bakersfield came from Topozone for which all map coordinates have been reprojected in NAD27. Suppose also that the distance from the center of Bakersfield to the furthest city limit is 2 mi.
Locality Type: Distance only
Coordinate Source: gazetteer
Coordinate Format: decimal degrees
Latitude (input): 35.373
Longitude (input): -119.018
Coordinate Precision: 0.001 degrees
Offset Distance: 5 mi
Measurement Error: 0
Distance Units: mi
Distance Precision: 1 mi
Decimal Latitude: 35.373
Decimal Longitude: -119.018
Uncertainty (m): 12213 m (7.589 mi)
Uncertainty sources: 143 m (0.089 mi) from coordinate precision 0.001 degrees, 3219 m (2 mi) from the extent of the named place, 8047 m (5 mi) from the offset distance, 805 m (0.5 mi) from the 1 mi distance precision

Distance Along a Path
Example: 13 mi E (by road) Bakersfield
Suppose the coordinates for this locality were interpolated to the nearest 1/10th minute from the printed USGS Taft 1:100,000 Quad map and the distance from the center of Bakersfield to the furthest city limit is 2 mi.
Locality Type: Distance along path
Coordinate Source: USGS map: 1:100,000
Coordinate Format: degrees decimal minutes
Latitude (input): 35 degrees 26.1' N
Longitude (input): 118 degrees 48.1' W
Coordinate Precision: 0.1 minutes
Measurement Error: 0.03107 mi (0.5 mm on 1:100,000 map)
Distance Units: mi
Distance Precision: 1 mi
Decimal Latitude: 35.435
Decimal Longitude: -118.8016667
Uncertainty (m): 4363 m (2.711 mi)
Uncertainty sources: 50 m (0.03107 mi) from printed USGS 1:100,000 map accuracy, 239 m (0.148 mi) from coordinate precision 0.1 minutes, 3219 m (2 mi) from the extent of the named place, 50 m (0.03107 mi) from the ability to distinguish 0.5 mm on a printed USGS 1:100,000 map, 805 m (0.5 mi) from the 1 mi distance precision

Distance Along Orthogonal Directions
Example: 2 mi E and 3 mi N of Bakersfield
Suppose the coordinates for Bakersfield (the named place) came from the GNIS database (a gazetteer), the coordinates of the locality were calculated to the nearest second, and the distance from the center of Bakersfield to the furthest city limit is 2 mi.
Locality Type: Distance along orthogonal directions
Coordinate Source: gazetteer
Coordinate Format: degrees minutes seconds
Latitude (input): 35 degrees 25' 4" N
Longitude (input): 118 degrees 58' 54" W
Datum: not recorded
Coordinate Precision: nearest second
North or South Offset Distance: 3 mi
North or South Offset Direction: N
East or West Offset Distance: 2 mi
East or West Offset Direction: E
Measurement Error: 0
Distance Units: mi
Distance Precision: 1 mi
Decimal Latitude: 35.4612939
Decimal Longitude: -118.946227
Uncertainty (m): 7441 m (4.624 mi)
Uncertainty sources: 3045 m (0.049 mi) from unknown datum, 40 m (0.025 mi) from coordinate precision nearest second, 3219 m (2 mi) from the extent of the named place, 1138 m (0.707 mi) from the 1 mi distance precision in two dimensions (the diagonal of a 0.5 mi by 0.5 mi square)

Example 1: 10 mi E (by air) Bakersfield
Suppose the coordinates for Bakersfield came from the GNIS database (a gazetteer), the coordinates of the locality were calculated to the nearest second, and the distance from the center of Bakersfield to the furthest city limit is 2 mi.
Locality Type: Distance at a heading
Coordinate Source: gazetteer
Coordinate Format: degrees minutes seconds
Latitude( input): 35 degrees 22' 24" N
Longitude(input): 118 degrees 50' 56 W
Datum: not recorded
Coordinate Precision: nearest second
Offset Distance: 10 mi
Measurement Error: 0
Distance Units: mi
Distance Precision: 10 mi
Decimal Latitude: 35.3733333
Decimal Longitude: -118.6717880
Uncertainty (m): 22208 m (13.8 mi)
Direction Precision: 45 degrees (between NE and SE, each 45 degrees from E)
Uncertainty sources: complicated combination of unknown datum, coordinate precision nearest second, extent of the named place, distance precision, and direction precision

Example 2: 10 mi ENE (by air) Bakersfield
Suppose the coordinates for the locality were interpolated to the nearest second from the USGS Gosford 1:24,000 Quad map on which you can distinguish between millimeters and the distance from the center of Bakersfield to the furthest city limit is 2 mi.
Locality Type: Distance at a heading
Coordinate Source: USGS map: 1:24,000
Coordinate Format: degrees minutes seconds
Latitude (input): 35 degrees 24' 21" N
Longitude (input): 118 degrees 51' 25" W
Coordinate Precision: nearest second
Offset Distance: 10 mi
Measurement Error: 0.007 mi (0.5 mm on 1:24,000 map)
Distance Units: mi
Distance Precision: 10 mi
Decimal Latitude: 35.4613445
Decimal Longitude: -118.6932627
Uncertainty (m): 12055 m (7.491 mi)
Direction Precision: 11.25 degrees either side of ENE
Uncertainty sources: complicated combination of map accuracy, unknown datum, coordinate precision nearest second, extent of the named place, measurement accuracy (0.5 mm at 1:24,000), distance precision, and direction precision

Preferred citation:
Wieczorek J (2001) MaNIS/HerpNet/ORNIS Georeferencing Guidelines. Museum of Vertebrate Zoology, University of California, Berkeley. Available: http://georeferencing.org/docs/GeorefGuide.html

1. NIMA, 2000. Department of Defense World Geodetic System 1984. Its Definition and relationships with local geodetic systems. TR8350.2, Third Edition, 3 Jan 2000. This document can be found at the following URL:

http://earth-info.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf
2. USGS, 1999. National Mapping Program Technical Instructions. Part 2. Specifications. Standards for Digital Line Graphs. This document can be found at the following URL:
http://nationalmap.gov/standards/pdf/Pdlg0999.pdf
3. NOAA. Differential GPS (DGPS) & Your Chart. Specified Chart Accuracy.
http://www.nauticalcharts.noaa.gov/nsd/DGPSchart.html
4. The results of this work were published as the point-radius method in the following manuscripts:

Wieczorek J, Hijmans R, Guo Q (2004) Point-radius method for georeferencing locality descriptions and calculating associated uncertainty. International Journal of Geographical Information Science 18(8):745-767. Available: http://georeferencing.org/docs/Wieczoreketal2004PointRadius.pdf

Chapman AD, Wieczorek J (eds.) (2006) Guide to Best Practices for Georeferencing. Copenhagen: Global Biodiversity Information Facility. Available: http://www.gbif.org/resource/80536

Zermoglio PF, Chapman AD, Wieczorek JR, Luna MC, Bloom DA 2020. Georeferencing Quick Reference Guide Available: http://georeferencing.org/docs/GeoreferencingQuickReferenceGuide.pdf

Chapman AD, Wieczorek J (eds.) (2020) Georeferencing Best Practices. Copenhagen: Global Biodiversity Information Facility. Available: http://www.gbif.org/resource/80536

Georeferencing Calculator:

Wieczorek C, J Wieczorek (2019) Georeferencing Calculator. Rauthiflor LLC. Available: http://georeferencing.org/georefcalculator/gc.html

Map Accuracy Standards:

http://nationalmap.gov/standards/nmas.html Township, Range, Section Information: TRS Graphical Locator
Township and Range - Public Land Survey System on Google Earth
National Atlas PLSS article