Earth Coordinates

&

Grid Coordinate Systems

You will always find parallels and meridians on large-

scale maps

This is done to make the map a very close

approximation to the size and shape of the piece of

the ellipsoidal earth that it represents

Horizontal Reference Datums

Datums are the collection of very accurate control

points (points of known accuracy) surveyors use to

geo-reference all other map data

Surveyors determine the precise latitude and

longitude of horizontal control points spread across

the landscape

Horizontal control point monuments are fixed

objects established by surveyors when they

determine the exact location of a point

See Figure 1.15, page 17, Muehrcke

Often found on top of a hill or other prominent

feature

From the 1920s to the early 1980s,these control

points were surveyed relative to the surface of the

Clarke 1866 ellipsoid, together forming what is

called the North American Datum of 1927 (NAD 27)

Topographic maps, nautical and aeronautical charts

and many other large-scale maps of this time period

had graticule lines or ticks based on this datum

See Figure 1.16, page 17, Muehrcke

The Corvallis, Oregon map was published in 1969

and the bottom-right corner of the map has the NAD

27 latitude and longitude

By the early 1980s, better knowledge of the earth’s

shape and size and far better surveying methods let

to the creation of a new horizontal reference datum,

the North American Datum of 1983 (NAD 83)

The NAD 27 control points were corrected for

surveying errors where required, then these were

added to thousands of more recently acquired

points

The latitudes and longitudes of all of these points

were determined relative to the Geodetic Reference

System of 1980 (GRS 80) ellipsoid, which is

essentially identical to the WGS 84 ellipsoid

The change of horizontal reference datum meant

that the coordinates for control points across the

continent changed slightly.

This change had to be shown on large-scale maps

published earlier but still in use

See Figure 1.16, page 17, Muehrcke

On topographic maps the new position of the map

corner is shown by a dashed “plus” sign

Many times the shift is in the 100 meter range and

must be taken into account when plotting on older

maps the latitudes and longitudes obtained from

GPS receivers

Even today, many countries continue to use local

datums or ellipsoids resulting in incompatibilities

The European Terrestrial Reference System of 1989

(ETRS 89) is a way to standardize mapping

coordinates but many countries continue to use

locally-derived ellipsoids

The Earth as a Geoid

When we treat the earth as an oblate ellipsoid, we

neglect mountain ranges, ocean trenches and other

surface features

There is justification for doing this

The earth’s surface is truly smooth when we

compare the surface undulations to the 7,918 mile

diameter of the earth

The greatest relief variation is the approximately

12.3 mile difference between the summit of Mt.

Everest (29,035 feet) and the deepest point in the

Marianas Trench (36,192 feet)

This vertical difference is immense on our human

scale, bit only 1/640th of the earth’s diameter

It has been said that is the earth were a bowling ball,

it would be smoother than the bowling ball

The top of Mt. Everest is located at:

o 27o59’N, 86o56’E, 29,035 feet (8,852 meters)

What is this elevation relative to?

Vertical Reference Datums

Elevations and depths are measured relative to what

is called a vertical reference datum

An arbitrary surface with an elevation of zero

The traditional datum used for land elevation is

mean sea level (MSL)

Surveyors define mean sea level as the average for

all low and high tides at a particular starting location

over a metonic cycle (the 19 year cycle of the lunar

phases and days of the year)

Early surveyors chose this datum because of the

measurement technology of the day

Surveyors first used the method of leveling, where

elevations are determined relative to the point

where mean sea level is defined

They used horizontally aligned telescopes and

vertically aligned leveling rods

A small circular monument was placed in the ground

at each surveyed benchmark elevation point

A benchmark is a permanent monument that

establishes the exact elevation of a place

Later, surveyors could determine elevation by

making gravity measurements at different locations

on the landform and relating them to the strength of

gravity at the point used to define MSL

Gravity differences translate into elevation

differences

Mean sea level is easy to determine along

coastlines, but what about inland locations?

What is needed is to extend MSL across the land

Imagine that the MSL is extended under the

continental land masses

This is the same as extending a surface having the

same strength of gravity as mean sea level

See Figure 1.17, page 18, Muehrcke

This imaginary equal gravity surface doesn’t form a

perfect ellipsoid because differences in topography

and earth density affect gravity’s pull at different

locations

The slightly undulating, nearly ellipsoidal surface

that best fits mean sea level for all the earth’s

surface is called a global geoid

The global geoid rises and falls approximately 100

meters above and below the oblate ellipsoid surface

in an irregular pattern

World maps showing land topography and ocean

bathymetry use land heights and water depths

relative to the global geoid surface

The mean sea level datum based on the geoid is so

convenient that it is used to determine elevations

around the world

It is used as the base for the elevation data found on

nearly all topographic maps and nautical charts

Be aware that the local geoid used in your area is

probably slightly above or below (usually within 2

meters) the global geoid elevations used on world

maps

This difference is caused by mean sea level at one or

more locations being used as the vertical reference

datum for your nation or continent, not the average

sea level for all oceans

In the US you may see elevations relative to the

National Geodetic Vertical Datum of 1929 (NGVD

29) on older topographic maps

This datum was defined by the observed heights of

mean sea level at 26 tide gauges, 21 in the US and 5

in Canada

It was also defined by the set of elevations of all

benchmarks resulting from over 60,000 miles of

leveling across the continent totaling over 500,000

vertical control points

North American Vertical Datum of 1988 (NAVD 88):

In the 1980s surveyors adjusted the 1929 datum

with new data

Mean sea level for the continent was defined at one

tidal station on the St. Lawrence River at Rimouski,

Quebec, Canada

NAVD 88 was a necessary update of the 1929 datum

since about 400,000 miles of leveling was added to

the NGVD since 1929

Additionally, numerous benchmarks had been lost

over the decades and the elevations of others had

been affected by vertical changes caused by rising of

land elevations since the retreat of glaciers at the

end of the last ice age (isostatic rebound) or

subsidence from sedimentation and the extraction

of natural resources like oil and water

GPS has created a second option for measuring

elevation

GPS receivers calculate what is called the ellipsoidal

height, the distance above or below the surface of

the WGS 84 ellipsoid along a line from the surface to

the center of the earth

See Figure 1.17, page 18, Muehrcke

An ellipsoidal height is not an elevation, since it is

not measured relative to the mean sea level datum

for your local geoid

You must convert GPS ellipsoidal height values to

mean sea level datum elevations before you can use

them with existing maps

See Figure 1.18, page 19, Muehrcke

Grid Coordinate Systems

There are several ways to pinpoint locations on a

map

The latitude and longitude graticule has been used

for over 2000 years as the worldwide locational

reference system

Geocentric latitude and longitude coordinates on a

sphere or geodetic latitudes and longitudes on an

oblate ellipsoid, still key to modern position finding,

are not as well suited for making measurements of

length, direction and area on the earth’s surface

The basic difficulty is the fact that latitude-longitude

is a coordinate system giving positions on a rounded

surface

It would be much simpler if we could designate

location on a flat surface using horizontal and

vertical lines spaced at regular intervals to form a

square grid

We could then simply read coordinates from the

square grid of intersecting straight lines

Most maps are created by projecting the earth’s

surface onto a flat surface

The advantage of the flat map projection surface is

that we can locate something by using a two-axis

coordinate reference system

This coordinate system is the basis for the square

grid of horizontal and vertical lines on a map

We call a plane-rectangular coordinate system

based upon and mathematically placed on a map

projection a grid coordinate system

To devise such a system for large areas, we have to

deal somehow with the earth’s curvature

We know that transferring something a spherical to

something flat always introduces geometrical

distortion

We also know that map projection distortion caused

by the earth’s ellipsoid shape is minimal for fairly

small regions

If we superimpose a square grid onto flat maps of

small areas, we can achieve positional accuracy good

enough for many map uses

All geographic grid systems are based on Cartesian

coordinates, invented in 1637 by the famous French

philosopher and mathematician Rene Descartes

Cartesian Coordinates

If you superimpose a square grid on a map, with

divisions on a horizontal x-axis and a vertical y-axis

where the axes cross at the system’s origin, you have

established the Cartesian Coordinate System

See Figure 4.1, page 64, Muehrcke

You can now pinpoint any location on the map

precisely and objectively by giving its two

coordinates (x,y)

The Cartesian Coordinate System is divided into four

quadrants (I-IV) based on whether the values along

the x- and y-axes are positive or negative

Mapmakers use only quadrant I for grid coordinate

systems so that all coordinates will be positive

numbers relative to the (0,0) grid origin

Grid Coordinates

The simplest way of defining map positions based on

Cartesian Coordinates has definite advantages over

using the spherical graticule to define positions

Measuring x- and y-coordinates from horizontal and

vertical axes with equally spaced distance

increments greatly simplifies locating environmental

features because you do not have to deal with the

decreasing separation between meridians

converging towards the poles

Grid coordinates systems based on the Cartesian

Coordinate System are especially handy for such

map analysis procedures as finding the distance or

direction between locations or determining the area

of a mapped feature like a lake

Two popular grid coordinate systems are:

o Universal Transverse Mercator (UTM)

o State Plane Coordinate System

Universal Transverse Mercator System

A grid coordinate system can be used worldwide if

enough zones are defined to insure reasonable

geometric accuracy

The best known is UTM

The UTM grid extends around the world from 84oN

to 80oS

60 north-south zones are used, each 6o longitude

See Figure 4.2, page 65, Muehrcke

Each zone has its own central meridian and uses a

secant case transverse Mercator projection

centered on the zone’s central meridian for each of

the 60 zones

This projection makes it possible to achieve a

geometrical accuracy of one part in 2,500 maximum

scale error

Scale factors ranging from 0.9996 to 1.0004 within

each zone

Each zone is individually numbered from west to

east

Each zone has separate origins for the northern and

southern hemispheres

Easting: the east-west x-coordinate in a grid

coordinate system. That is the distance east from the

origin

In both the northern and southern hemispheres, an

easting value of 500,000 meters (written

500,000mE) is assigned to the central meridian of

each UTM zone

This value, called the false easting, is added to all x-

coordinates so that there are no negative eastings in

the zone

A northing is the north-south y-coordinate in a grid

coordinate system

In the northern hemisphere a northing value of 0mN

is assigned to the equator so all northing values are

positive numbers

In the southern hemisphere, the equator is given a

false northing of 10,000,000mN

There are no negative y-values in the southern UTM

zone because this false northing values places the

origin of the zone very close to the south pole

UTM Example

See Figure 4.3, page 66, Meuhrcke

Zone 10 covers much of the western seaboard

The x-axis follows the equator

The central meridian for the zone is 123oW

The longitude range is 120oW to 126oW

The origin lies on the equator 500,000 meters west

of the central meridian at 123oW

UTM Coordinates of the 68th St Entrance to the North

Building

587390.92mE, 4513546.43mN, Zone 18 North

The near global extent of the UTM grid makes is a

valuable worldwide referencing system

The UTM grid is indicated on many foreign maps and

on all recent USGS maps

All GPS vendors program the UTM specifications

into their receivers

Note that UTM coordinates will differ when different

datums are used

You should check the datum information in the GPS

receiver to assure that the coordinates are being

recorded in the correct system

Because meridians and not stat boundaries delimit

UTM zones, it usually takes more than one UTM

zone to cover a state completely

Universal Polar Stereographic System

UTM grid zones extend from 80oS to 84oN

To complete the global coverage, a complementary

rectangle coordinate system called the universal

polar stereographic (UPS) system was created

UPS grid consists of a north zone and a south zone

Each zone is superimposed on a secant case polar

stereographic projection that covers a circular

region over each pole

See Figure 4.4, page 66, Muehckre

The north zone extends from 84oN to the north pole

The UPS coordinate at grid center is:

o 2,000000mE, 2,000,000mN

This assures all coordinates are positive

Virtually all large-scale maps of these high latitudes

are based on the UPS grid

The UTM grid system was not extended to the poles

because the 60 zones converge at the poles meaning

that a new zone would be encountered every few

miles

State Plane Coordinate (SPC) System

The state plane coordinate system was created in

the 1930s by the land surveying profession in the US

as a way to define property boundaries that would

simplify computation of land parcel perimeters and

areas

The idea was to completely cover the US and its

territories with grids laid over map projection

surfaces so that the maximum scale distortion error

would not exceed 1 part in 10,000

A distance measured over a 10,000 foot course

would be accurate to within a foot of the true

distance

This level of accuracy could not be achieved if only

one grid covered the whole country, because the

area is too large

The solution was to divide each state into one or

more zones and make a separate grid for each zone

The US was originally divided into 125 zones

Each has its own projection surface based on the

Clarke 1866 ellipsoid and NAD 27 geodetic latitudes

and longitudes

Most states have several zones

See Figure 4.5, page 67

Secant case Lambert conformal conic projections

are used for states of predominantly east-west

extent

Secant case transverse Mercator projections are

used for states of greater north-south extent

States with more than one zone use the names

North, South, East, West and Central to identify

zones

California uses Roman numerals

In recent years the following states have combined

zones into a single one:

o Nebraska

o South Carolina

o Montana

The logic of the SPC system is simple

Zone boundaries follow state and county

boundaries because surveyors have to register land

surveys in a particular county

Each zone has its own central meridian that defines

the vertical axis for the zone

An origin is established to the west and south of the

zone

Usually 2,000,000 feet west of the central meridian

for the Lambert conformal conic zones and

500,000 feet for the transverse Mercator zones

This means that the central meridians will usually

have an x-coordinate of either 500,000 feet for the

transverse Mercator or 2,000,000 for the Lambert

conformal conic zones

These large numbers for zone centers were selected

so that all x-coordinates will be positive numbers

Although different for each zone, the origin is always

at a parallel to the south of the zone to ensure that

all y-coordinates will be positive numbers

SPC coordinates are read in the same manner as

UTM coordinates – first to the east and then to the

north of the zone’s origin

For example the location of the state capitol dome in

Madison Wisconsin is:

o 2,164,600 ft E, 392,280 ft N, Wisconsin, south

zone

In 1983 the SPC system was modernized by

switching to NAD 83 and the GRS 80 ellipsoid

Zones were redefined in metric units

The SPC system served the needs of states when it

was created and state plane coordinates have been

widely used for public works and land surveys

The SPC system is now largely obsolete as far as

surveyors are concerned

One reason is modern surveying methods can

exceed the accuracy

Also, each SPC zone is a separate entity with its own

grid definition – a fact that frustrate and discourages

uses across zone boundaries

Other Grid Systems

US state grids – combine UTM zones into a single

state grid

European grid coordinate systems

British National Grid coordinate system

Swiss coordinate system