WHERE THEORY MEETS PRACTICE

Transcription

1 world from others, leica geosystems WHERE THEORY MEETS PRACTICE A NEW BULLETIN COLUMN BY CHARLES GHILANI ON PRACTICAL ASPECTS OF SURVEYING WITH A THEORETICAL SLANT february 2012 ² ACSM BULLETIN ² 27

2 USGS A COMMON REFERENCE FOR GIS MAPPING : THE STATE PLANE COORDINATE SYSTEM This is the first in a series of articles on state plane computation commonly needed by practitioners but not always understood. BY CHARLES D. GHILANI The ellipsoid is a mathematical surface on which all points on the Earth can be assigned geodetic coordinate values of latitude and longitude. Map projections establish a 1-to-1 relationship between points on the ellipsoid and those on the map projection surface. The process of taking a curved surface and placing it on a flat surface will always introduce some distortions to the projected objects. A well known distortion present on many maps is that areas are enlarged at the 28 µ extremities of the projection. For example, Greenland appears much larger on a map using a Mercator projection than it does on a globe of the Earth due to the process of transforming the geodetic coordinates of the boundary to their equivalent map coordinates. All map projection systems introduce some distortions when they are applied to the Earth. For example, some will result in scaling differences between distances while others will distort shapes of objects. A conformal ACSM BULLETIN ² february 2012 map projection preserves shapes of objects at the expense of distances. Preserving shapes means that angles are preserved for a limited region about a point. These particular projections are convenient for surveying applications since there is little need to reduce angles. However there is an exception to this when the sight distances become long. The map projection systems used for state plane coordinate systems have scaling differences in distances and

3 theory & practice areas primarily. To maintain the accuracy of your survey when using a state plane coordinate system, the scaling differences between distances measured at the surface of the Earth with those on the map projection surface must be taken into account. Figure 1 depicts the differences between horizontal distances measured at the surface of the Earth L m and the equivalent geodetic length of L e and mapping surface length of L g. When the National Geodetic Survey established the first state plane coordinate system (SPCS) zone in the 1930s, it was decided to limit the scaling differences between geodetic distances on the ellipsoid and their equivalent map (grid) distances to be better than 1:10,000. This meant that a survey performed on the ellipsoid (near sea level in the 1930s) would not have to reduce any measured distances to maintain 1:10,000 survey accuracies. This worked fine for the 1930 surveys where typical accuracies were well under 1:10,000. However as can be seen in Figure 1, if the survey is performed at some elevation, the scaling differences between the measured distance and the geodetic distance can be significant due to the elevation of the line. In fact, at 2300 ft, the elevation factor will yield results that are less than 1:10,000. Figure 1. Differences between the measured distance L m, geodetic distance L e, and the map distance L g. Today, surveys performed at less than 1:10,000 would be considered deficient. Thus proper reductions must be performed to maintain the accuracies of the survey no matter where the survey is performed. Additionally, the vertical datum (geoid) is about 100 ft below the ellipsoid throughout the conterminous United States. What does this mean? This means that all observed horizontal distances must be reduced to account for both the geodetic height of the observing station and scaling differences between the geodetic and mapping distances. Grid versus Observed Distances Geodetic distances can be determined from observed horizontal distances by multiplying the measured distance with an elevation factor. The elevation factor is given by following equation. Re Re EF R h R H N e where R e is the radius of the Earth, h is the geodetic height at the observing station, H is the orthometric height (commonly known as the elevation) at the observing station, and N is the geoid height at the observing station. Since the mathematical surface of the Earth is best approximated by an ellipsoid, the radius of the Earth varies at any point, depending on the azimuth of the line. However, for all but the most precise surveys, the mean radius of the Earth (6,371,000 m or 20,902,000 ft) works sufficiently well. Once the distance is scaled to its equivalent geodetic distance, it can be further scaled to its equivalent mapping distance using the scale factor of the line. This scale factor is a product of the state plane coordinate computations and is often supplied by the conversion software during the computational process. The scale factor is a function of the latitude of the point. Because most survey lines are relatively short, an average of the scale factor at the end points of the line is typically used for the scale factor of the line. If, however, long lines are observed, NGS recommends a weighted scale factor be used for each line. This equation is: where k is the scale factor for the line, k 1 is the scale factor at the observing station, k 2 is the scale factor at the e february 2012 ACSM BULLETIN 29

4 theory & practice sighted station, and k mid is the scale factor at the midpoint of the line. As most surveys are limited in area, it is often possible to establish a single combined factor for the entire survey. This combined factor is the product of an average elevation factor for the survey and an average scale factor. All of today s survey controller software allows the user to input this single combined factor into their software. When this is done, the software reduces all observed horizontal distances to their map equivalents before computing coordinates and applies the inverse when staking out coordinates. When can a single combined factor be used in a survey? It depends on the intended accuracy of the survey, elevation difference in the survey, average elevation of the survey, and the length of the longest observed distance. For example, in most surveys in Pennsylvania, the lengths of lines are limited by environmental conditions; in general, they are less than 1000 ft. This means that the typical observed distance has five significant figures (###.## ft). A simplistic definition for a significant figure is all digits that are common plus one that varies. In other words, the distance is certain to 0.1 ft, but the 0.01 ft digit can vary with each observation. The values computed for the various factors should thus have at least four common digits plus one that varies. However, it would be better if the digits in the scale factors were not to vary in the places held by the number of significant figures in the distances, as that would ensure no variability in the elevation factors first five decimal places. For example,assume the average elevation of a survey is 300 m (~1000 ft) and that the survey varies in elevation by 30 m (~100 ft). Furthermore assume that the geoid height for the factors for lowest (285 m) and highest (315 m) points in the survey are and , respectively. These factors vary in the fifth decimal place, matching the precision of the observed distance. Now assume that the survey was performed in the PA South zone at latitude , which yields a scale factor of Further assume that the survey goes about 2000 ft north to latitude , which has a scale factor of The significant figures for these scale factors vary in the fifth decimal place. If the lowest elevation is at , then the combined factor for this point is , which yields If the highest elevation is at , its combined factor is , which yields These combined factors have four common digits and a different digit in the fifth decimal place, which yields five significant figures. For observed length of ft, the reduced map lengths using the two combined factors will be ft and ft. This leads to a difference of only ft. In fact, for a ft line the difference is only 0.01 ft. If this difference is more than can be tolerated for the purpose of the survey, then full reductions should be performed for every line in the survey. Alternatively, an average combined factor can be determined for the survey. In this case it would be Using the average combined factor, a ft line would correspond to ft of its true map length. This single combined factor can be entered into a data collector which will then perform the necessary reductions for computation of coordinates and stake out of lines. Grid versus Geodetic Azimuths In the state plane coordinate systems, all meridians on the map are parallel to the map's central meridian. Thus the direction of grid north at any location on the map is parallel to the central meridian. Since geodetic meridians all converge at the pole, azimuths of lines based on grid north will be different than azimuths based on geodetic north except at the central meridian. The difference in the azimuth is known as the convergence angle. Convergence angles east of the central meridian are considered positive and are considered negative west of the central meridian. From Figure 2 we conclude that the relationship between grid north and geodetic north is: Az grid = Az geodetic where Az grid is the grid azimuth of the line, Az geodetic is the equivalent geodetic azimuth, and is the convergence angle at the observing station. Figure 2. Difference between geodetic and mapping azimuths of a line. The convergence angle for any line can be computed as: = (77 45 )n where is a positive western longitude of the observation station, and n is a defined zone constant. If lines of sight are over 8 km, an additional secondterm (also known as the arc-to-chord) correction is needed. This correction is discussed in Chapter 20 of Elementary Surveying: An Introduction to Geomatics (Ghilani, 2012). 30 ACSM BULLETIN february 2012

5 theory & practice L-R: ohiolanndsurveys.com; mrsc.org; mssparky.com To properly compute coordinates in a state plane coordinate system zone, one must reduce all the distances using the appropriate combined factor. If a geodetic azimuth is available for the starting course, then its grid azimuth needs to be computed. If, however, there is a usable grid azimuth, no conversion is required. Summary The bottom line is that only map values should be used to compute mapping coordinates if you wish to compute state plane coordinates and maintain the accuracy of your survey. However, as we have seen, it is possible to minimize work involved in computing mapping coordinates by using a single combined factor in distance reductions. Once a project scale factor has been determined for the survey and entered into your data collector, all observed distances will be correctly scaled to their mapping equivalents which, in turn, will be used to correctly compute state plane coordinates. During stake out, the state plane coordinates help to determine the mapping length and then correctly scale these lengths to the surface for proper placement of the stake in the ground. Another application is in GIS (geographic information systems) mapping, where the state plane coordinate system provides a common reference plane to allow surveys of various types to be combined into a cohesive map. However, if state plane coordinates are incorrectly computed, the end result will be a mismatch of mapping elements that were located by various surveys and, in the end, a map of little or no value. ALTA / ACSM STANDARDS A regular column by GARY R. KENT, L.S., on all issues related to ALTA/ACSM Land Title Surveys The specific Table A items in the 2011 ALTA/ACSM Standards end with item 21. My question is regarding item 22, which is blank. Q: We have run into a lender who wants us to add a Table A item 22, stating measured finished floor elevations of structures on the property, and a Table A item 23, to include a note on the survey stating what the required parking would be (as determined by use, and by the zoning ordinance). Whether a surveyor should include a statement about required parking on a survey is subject to great debate. Our question is whether the ALTA/ACSM standards allow for the inclusion of multiple additional Table A items, and if so, what would be the best way to document this on the face of the survey? A : The committee did anticipate that there could be more than one additional item. But, any and all additional items need to be identified as Item 22, since there is no Item 23 provided for in the Standards. Thus, if there are more than one additional item, they should be identified as 22a, 22b, 22c, etc. Otherwise there will be confusion, as the Standards Reference only mention 22 Table A items. If the surveyor and client agree to any Ghilani, Charles D. and Paul R. Wolf additional items, the content of those items needs to be explained with Elementary Surveying: An Introduction a note, such as, in the case of your first example, Table A, item 22 perto Geomatics. Prentice Hall Publishers, tains to establishing finished floor elevations on the structures on this Upper Saddle River, NJ. property. Without any explanation, stating that you certified to Table A item 22 (or 22a and 22b, in the case of multiple items) would not An article with the same topic but different rendition mean anything to the person reviewing the survey. Gary Kent can be previously appeared in the Pennsylvania Surveyor. contacted at gkent@schneidercorp.com februaryy 2012 ² ACSM BULLETIN ² 31