Map Projections (1/2) Francisco Olivera, Ph.D., P.E. Center for Research in Water Resources University of Texas at Austin

Overview Geodetic Datum Map Projections Coordinate systems Global Positioning System

Definition A geodetic datum defines the size and shape of the earth, and the origin and orientation of the axis used to define the location of points. Over time, geodetic data have evolved from simple flat surfaces and spheres to complex ellipsoids. Flat earth models can be accurate over short distances (i.e., less than 10 Km), spherical earth models for approximate global distance calculations, and ellipsoidal earth models for accurate global distance calculations.

Shape of the Earth We think of the earth as a sphere when it is actually an ellipsoid, slightly larger in radius at the equator than at the poles.

P Oa b X   Ellipse Z An ellipse is defined by: Focal length =  Flattening ratio: f = (a-b)/a Distance F 1 -P-F 2 is constant for all points P on ellipse When  = 0 then ellipse = circle For the earth: Major axis: a = 6378 km Minor axis: b = 6357 km Flattening ratio: f = 1/300 F1F1 F2F2 P

Ellipsoid or Spheroid O X Z Y a a b Rotational axis Rotate an ellipse around one of its axis.

Standard Ellipsoids Ref: Snyder, Map Projections, A working manual, USGS Professional Paper 1395, p.12

Standard Horizontal Geodetic Data NAD27 (North American Datum of 1927) uses the Clarke (1866) ellipsoid. NAD83 (North American Datum of 1983) uses the GRS80 ellipsoid. WGS84 (World Geodetic System of 1984) uses GRS80.

Earth Surfaces Geoid is a surface of constant gravity. Topographic surface Ellipsoid Sea surface Geoid

Earth Surfaces Ocean Geoid Topographic surface Ellipsoid Gravity Anomaly

Elevation P z = z p z = 0 Mean Sea level = Geoid Topographic Surface Elevation is measured from the Geoid

Standard Vertical Geodetic Datum A vertical datum defines elevation z, taking into account a map of gravity anomalies between the ellipsoid and the geoid. NGVD29 (National Geodetic Vertical Datum of 1929). NAVD88 (North American Vertical Datum of 1988).

Overview Geodetic Datum Map Projections Coordinate systems Global Positioning System

Map Projections A map projection is a mathematical algorithm to transform locations defined on the curved surface of the earth into locations defined on the flat surface of a map.

Map Projection Representative Fraction Globe distance Earth distance Scale Projection (e.g. 1:24,000) (e.g ) Scale Fraction Map distance Globe distance

Types of Projections Conic: Screen is a conic surface. Lamp at the center of the earth. Examples: Albers Equal Area, Lambert Conformal Conic. Good for East-West land areas. Cylindrical: Screen is a cylindrical surface. Lamp at the center of the earth. Examples: (Transverse Mercator). Good for North- South land areas. Azimuthal: Screen is a flat surface tangent to the earth. Lamp at the center of the earth (gnomonic), at the other side of the earth (stereographic), or far from the earth (orthographic). Examples: Lambert Azimuthal Equal Area. Good for global views.

Conic Projections Albers and Lambert

Cylindrical Projections Transverse Oblique TangentSecant Mercator

Azimuthal Lambert

Albers Equal-Area Conic

Lambert Conformal Conic

Universal Transverse Mercator

Lambert Azimuthal Equal-Area

Distortion Projected Maps In the process of transforming a curved surface into a flat surface, some geometric properties are modified. The geometric properties that are modified are: Area (important for mass balances) Shape Direction Length The difference between map projections has to do with which geometric properties are modified. Depending on the type of analysis, preserving one geometric property might be more important that preserving other.