Before we start: Important changes TA change: Tony Randolph is no longer a TA for this course New TA is Paul McCall paulmc@email.unc.edu The syllabus has been changed ot reflect this New method of turning in assignments We’ll use the digital drop box on Blackboard This will be explained in more detail in your lab assignment

Geodesy

Locating Positions on the Earth Basic Geodesy Map Projections Coordinate Systems Scale Today we’ll talk about Geodesy, Coordinate systems and projections will come next week

Geographic Data Features must be referenced to some real world location Known as georeferencing In order to use geographic data, the features in the data must be referenced to some real world locations. This is known as georeferencing. Without georeferenced data, it is very difficult to perform any spatial analyses, because your different data sets will not overlay.

Geographic Data & Position GOAL: To assign a location to all features Important elements must agree: ellipsoid datum projection coordinate system scale The goal in georeferencing data is to have a real world location assigned to all features of your data. This will allow all comparable locations in different data sets to agree and overlay properly. Seems pretty straightforward, right? Well, there’s more to it than that. In order to use multiple data sets together, not only do they all have to be georeferenced, but there are several important elements of the data that must agree between all the data sets. These elements are ellipsoid, datum, projection, and coordinate system. Scale is also important, but more as a potential source of error. I will discuss scale more later.

Position/location To determine position on the Earth, it is necessary to understand how those elements relate to one another Begin with geodesy The relationship we’ll talk about today is the relationship between ellipsoid and datum

What is Geodesy? Geodesy: “A branch of applied mathematics which determines by observation and measurement the exact positions of points and the figures and areas of large portions of Earth's surface, the shape and size of the Earth, and the variations of terrestrial gravity." More complete definition of geodesy

What is Geodesy? More simply, geodesy is the study of the Earth’s size and shape. Simply put Geodesy is the study of the earth’s size and shape.

The Earth is Not Flat Piece of cake right? The earth is round everyone knows that. And has known it for a long time. Aristotle (384-322 B.C.) talked about the sphericity of the Earth, doubts remained until Magellan first circled the globe

Eratosthenes (276-196 B.C.) Measured the height of an obelisk in Alexandria and determined the circumference of the earth. His measurement: 25,000 Miles Modern Measurement: 24,860 Miles Era-tos-thenes He had heard of a well in Syene (near modern Aswan Egypt - Tropic of Cancer) where the sun’s reflection could be seen in the water at noon on June 21. -- The sun was directly overhead. At the exact moment the sun could be seen in the well (solar noon) he measured the length of the shadow cast by an obelisk in Alexandria. Then using some basic geometry he was then able to calculate the circumference of the earth by creating a triangle between Alexandria, Syene and the Sun. He got pretty close…

So the Earth is Round, Right? So everyone knows the Earth is round, even Columbus. Case closed right? Not exactly, the earth isn’t perfectly round either. What shape is it?

The Earth is Irregular Not perfectly round due to: Distortion due to the Earth’s rotation Irregularities due to variations in gravity Small irregularities on the surface such as mountains, basins, etc. The earth is not completely round, there are irregularities in the shape of the earth due to:

The Earth is Irregular Slightly flattened at the poles Equator bulges Southern Hemisphere slightly larger than Northern Hemisphere So if the earth is irregularly shaped, how do we represent that shape? How do we account for the irregularities? We can think of the earth in three different ways…

The Earth is: A Spheroid An Ellipsoid A Geoid A spheroid is the simplest, but least accurate of the 3. In general this is OK for very small scale maps (i.e., very large areas like the globe or continents), OR at large scales (very small areas) where distortion is likely to be fairly uniform and very small relative to the short distances. BUT at intermediate scales like counties, states, regions, etc. this does matter. An ellipsoid is most commonly used because it is relatively simple mathematically, but more accurate than a spheroid. A geoid is the most accurate, but also the most complicated approximation of the earth’s size and shape. Geoids certainly exist and are used occasionally, much less often than ellipsoids or spheroids

The Earth as a Sphere Geographic coordinates (latitude/longitude) used to specify locations. Treating the Earth as a sphere is accurate enough for small maps of large areas of the Earth (i.e. very small scale maps) If you portray the Earth as a sphere latitude and longitude are used to specify locations. As long as your scale is less than 1:100,000 a spheroid can be used. 1:1,000,000 OK 1:50,000 Not OK 1:1000 OK again Another consideration is how accurate you NEED your data to be. (Lat/long was initially a spherical coordinate system -- Ptolemy who invented Lat/Long assumed world was Spherical)

The Earth as a Sphere Spheroid model: short range navigation global distance approximations The slight flattening at the poles result in a 20 km difference at the poles from the average spherical radius We can use the spheroid for short range navigation, because at short ranges, the surface is effectively spherical. We can also use this model for global distance approximations. However, the spheroid does not accurately model the shape of the earth.

The Earth as an Ellipsoid Ellipsoid is a flattened sphere Ellipsoid is created by rotating a 2 -dimensional ellipse around an axis. As we mentioned earlier, the Earth isn’t a perfect sphere. Because of the bulging its really an ellipsoid. When you rotate an ellipse around an axis that’s known as an ellipsoid. (That’s the difference between an ellipse and ellipsoid. An ellipsoid is a rotating ellipse)

The Earth as an Ellipsoid Every ellipsoid has a semi-major (a) and semi-minor axis (b) The amount of flattening is defined as a value f, which is calculated using the semi-axes

The Earth as an Ellipsoid a = semi-major axis b= semi-minor axis f = ((a-b)/a) = flattening

The Earth as an Ellipsoid: WGS84 Ellipsoid b a Let’s take that same formula and transfer it to the earth. In this example, we are looking at the WGS84 ellipsoid. There are different surveys of the earth that use different values for a and b but will settle on f equaling 1/298.257223562 or 0.003353 f = 0.003353

The Earth as an Ellipsoid Here are some other commonly used ellipsoids. You don’t need to memorize any of this. Just keep in mind the term reference ellipsoid. That’s what an ellipsoid is used for… reference. It’s a starting point to calculating a position on the earth. Do note the dates. This is a science that has evolved through time as our ability to measure the earth has improved. Therefore the more recent ellipsoid calculations (e.g., WGS 84 and GRS 1980) are more accurate than those that came before.

Differences in Lat / Long On a spheroid, lines of latitude (parallels) are equal distance apart On an ellipsoid, the distance between parallels slightly increases as latitude increases On a spheroid, the length of one degree of latitude is equal from pole to pole. But, on an ellipsoid, the flattening makes the length of a degree increase as you near the poles. This is something to keep in mind as I continue on in this lecture.

The Earth as a Geoid The only thing shaped like the Earth is the Earth Geoid means “Earth Like” Shape is based on gravity field corrected by the centrifugal force of the earth’s rotation. So we’ve seen the earth as a spheroid and an ellipsoid, but all of these are approximations because nothing is shaped like the earth, except the earth. So scientists came up with an object that is truly earth shaped -- a geoid. Gravity is more important than mountains because even the distance from the highest mountain to the bottom of the ocean is only about 20 km, compared to the diameter of the earth which is over 12,750 km. This is a very small fraction. Gravitational attraction (as I’m sure we all remember from physics) relates to the mass, and therefore the density, of an object. Since the earth is made of different materials, gravity affects different areas differently, and can lead to some bulging in the shape of the earth. Density is important, for example, because continental crust (land on earth) is less dense than oceanic crust. Also remember that most of the mass of the earth is not at the surface, so the density of materials in the core & mantle are even more important than the crust.

The Earth as a Geoid Geoid -- The surface on which gravity is the same as its strength at mean sea level Coincides with the surface to which the oceans would conform over the entire earth if it were made only of water.

The Earth as Geoid If the Earth were completely uniform in its geological composition, landforms and density, then the geoid would match the ellipsoid exactly But the earth is not completely uniform. So we can use a geoid to model the earth’s surface instead of an ellipsoid. And surveyors DO use the geoid to make precise locational calculations. But for our purposes, and for most purposes in GIS, we don’t use the geoid. It’s too complex. Instead, we use an ellipsoidal model.

Interaction Spheroid Ellipsoid Geoid So now we have a spheroid, which is the simplest model of the shape of the earth. We have an ellipsoid, which is a much more accurate model of the shape of the earth. And we have this complex, irregularly-shaped geoid, which is the most accurate model for the shape of the earth. How can we get these three elements to work together? How do we know where locations in a geographic coordinate system (lat/long) are, relative to each of them: spheroid, ellipsoid and geoid? We’ve seen that the calculation of latitude is different between spheroids and ellipsoids. Trying to calculate it on an irregular geoid is very mathematically complex, and would be nearly impossible without an advanced degree in math. Somehow, we need to find a simpler way of getting an accurate location.

Datums Datum -- n. (dat - m) \ any numerical or geometric quantity which serves as a reference or base for other quantities e So we have datums. Now, I said we need a simpler way, and simple is a relative term. Datums are complex models in and of themselves. But the key to the Geodetic Datum is that it provides a link between locations on the sphere ellipsoid and geoid. Basically a Datum is a why of linking the irregular surface of the Earth to a reference ellipsoid. The linking begins as a point of origin, and accuracy (e.g., the relationship between the modeled ellipsoid or sphere and the actual surface of the earth) decreases the further from the origin we get. Datums are linked to the ellipsoids they are based upon, so when we choose to project our data to a Datum (e.g., NAD 83) we are also choosing an elipsoid.

Geodetic Datum Geodetic datum – The information that ties an ellipsoid model to the geoid model Horizontal datum (most commonly used) Vertical datum Geodetic Datum is used as a reference base for mapping. Datums can be horizontal or vertical. We will mostly be talking about horizontal datums. Datums are also tied to a reference ellipsoid.

Horizontal Datum Components Parameters of the ellipsoid axis length flattening value Parameters that tie the ellipsoid to the origin point (known place on the Earth) So to define a geodetic datum you need an origin point. This is a place where everything will be referenced from. Until recently, all datums were regional in nature. Why do you think that a datum would be regional? Datums are regional because the complexity of the geoid model makes accurate calculations difficult over large differences when you tie a datum to one specific location on the surface. For instance for the NAD27 datum, the origin point was a place called “Mead’s Ranch” in North Central Kansas. NAD27 stands for North America Datum 1927, and until the 80s it was THE reference datum for North America.

Geodetic Control Networks Geodesists and surveyors create geodetic control networks to precisely link a set of known locations to each other and to the ellipsoid/geoid at a datum origin. Create Geodetic Control Networks Horizontal control networks (expressed in lat/long) and vertical control networks (expressed in lat/long and elevation)

Geodetic Control Networks Surveyors use these known points in the control networks for surveys and mapping. Control Points are referenced to specific datums.

Geodetic Control Network How do surveyors and others know where these points are?

Geodetic Control Network They mark it. Over 800,000 points in the U.S. Sundial -- EZ1375 Keep your eyes out for others as you walk around campus.

Common Ellipsoids, Datums I mentioned a few minutes ago that datums are tied to specific ellipsoids. Well, here are some commonly used datums and ellipsoids. I also said that datums are regional. You can see here that the NAD 1927 datum and it’s Clarke 1866 ellipsoid are used for North America. Now, there are also global datums. These are also called topocentric datums. Topocentric datums have the center of mass of the earth as their reference point, and can be used to calculate positions across the entire planet. One example of a topocentric datum is WGS84, or World Geodetic System 1984. It is probably the most commonly used datum today. It is tied to an ellipsoid of the same name. It is also the datum and ellipsoid that the Global Positioning System uses to calculate positions.

Geodetic Datums The purpose of all of this is to end up with a very, very accurate map. Not all maps need this accuracy, but some do.

Geodetic Datums If Datums are consistent you can end up with an inaccurate map. Geographer Peter Dana at Univ. of Texas calculated the shift of the marker at the Texas Capitol. Now, this gets us back to one of the points I made at the beginning of class. When using several data sets in a GIS, it is important that they all have the same datum. Otherwise, there will be an offset between your data sets, and it is all due to the datum. Now, it is also important that the ellipsoids be the same, but since each datum has a specific ellipsoid, that sort of goes without saying. And it’s also important to choose a datum that is applicable to your study area. Since many are regional, it doesn’t make sense to use, for instance, the Tokyo datum to define your positions in North Carolina.

Review Sphere – The simplest 3D model of the earth Ellipsoid – A more accurate model that takes into account some of the Earth’s irregularity Geoid – The most accurate, most complex model of the Earth, taking into account the Earth’s minor variations from an ellipsoid Datum – The information that ties an ellipsoid model to a known place on the Earth