2018 GIS in the Rockies Workshop: Coordinate Systems and Projections

COORDINATE SYSTEMS &
PROJECTIONS:
John Hunter, PLS
Colorado NGS Geomatics Coordinator
Grid v. Ground

Interview of Dave Doyle, Chief Geodetic Surveyor, NGS, Ret. North Carolina Professional Land Surveyors Society (2014)

• Coordinate System Basics
• Ellipsoids
• Projection Basics
• Distortion
• Horizontal Datums
• Heights & Scale Factors
• Localized Ground Projection
• Grid v. Ground
• SPCS 2022
The Stuff We Are Going To Learn About
9/25/20183

Geocentric Geodetic Plane
Types of Coordinates Systems
9/25/20185

Types of Coordinates Systems
9/25/20186
Tertiary
Positions
Secondary
Positions
Primary
Positions
Geocentric
Geodetic
Projected

Types of Coordinates
and Respective Coordinate System
9/25/20187
Coordinate Type Coordinate System Note
Earth Centered, Earth Fixed
P(x,y,z)
Geocentric GPS/GNSS Points
Traditional Cartesian

Earth-Centered-Earth-Fixed Coordinates are the most
common Geocentric.
Geocentric coordinates can be thought of as a 3 dimensional
Cartesian with X, Y, and Z axis.
The X axis:
From the origin extending, horizontally, through the
intersection of the Prime Meridian, and the Equator forming a
90° angle with the Z axis in the vertical plane.
The Y axis:
From the origin extending, horizontally, through the Equator
forming a 90° angle with the X axis in the horizontal plane,
and a 90° angle with the Z axis in the vertical plane.
The Z axis:
From the origin (center of the ellipsoid, or center of the earth)
extending straight through the north pole is the Z axis (not the
same as elevation).
GNSS/GPS surveyors are very familiar with these types of
coordinates as these are the results of raw GNSS/GPS
observations (our controllers take those and display them
into something we want to look at such as N, E, El).
GNSS/GPS ECEF are not influenced by ellipsoids, geoids,
projections, scale factors, or anything of the like, other than
the center point of the earth.
Geocentric Coordinate Systems
9/25/20188
ge·o·cen·tric
Geocentric:
Having or representing the earth as the center, as in former
astronomical systems.
Astronomy
Measured from or considered in relation to the center of the earth.

9/25/20189
Geocentric Coordinate Systems
Z
X
Y
P(X,Y,Z)
 Foundational or
Fundamental Positions.
 3D Cartesian Coordinates.
 Uses the center of the
earth as the origin (0,0,0)
of the coordinate system.
 Positions are not affected
by the size or shape of the
earth other than the earth
center of mass.

9/25/201810
Latitude & Longitude
DD°MM’SS.SSSS”N, DDD°MMM’SS.SSSS”W
Geodetic Differs with Ellipsoid
GRS80 Vs. Clarks

Traditional World Coordinate Systems
Considering the same Cartesian system with X, Y, and
Z axis, the Geodetic coordinates utilize a measure of
angles to compute positions.
Latitude (geodetic) is the angle of measurement from
the equator to the point being observed, THAT IS
PERPENDICULAR TO THE ELLIPSOID. This line
does not extend through the origin except at the poles
and along the equatorial plane. As in not geocentric.
Because the constraining factor in latitude is that it is
perpendicular to the size and shape of the earth,
makes it geodetic.
Longitude is the angle measured AROUND the Z axis,
from the Prime Meridian to the point of observation.
This angle is measured counter clock-wise
The common misconception among GNSS/GPS
surveyors are that these are the coordinates of
GNSS/GPS and have become synonymous with
WGS84 coordinates. GRS80 and WGS84 are
essentially the same.
The truth is that geodetic coordinates such as Lat Long
are calculated by the GNSS/GPS controller from ECEF
coordinates.
Because these points are on the ELLIPSOIDAL
surface, the issue of heights, and elevations must be
accounted for.
Geodetic Coordinate Systems
9/25/201811
ge·o·det·ic
Of or relating to geodesy, especially as applied to land
surveying.
ge·od·e·sy
The science dealing with the shape and size of the earth
or large portions of it.

9/25/201812
Geodetic Coordinate Systems
b
a
a
P(L,L,h)
 Secondary Positions.
 Angular Coordinates.
 Uses the same X,Y,Z
geocentric reference frame
for the origin.
 Positions are dependent
upon the size or shape of the
earth.
 Positions are based upon the
mathematical approximation
of the earth.
Greenwich
b

Northing and Easting Coordinates are useful for
representing geospatial information on a flat surface.
Commonly used in most Surveying and Engineering
applications.
2D cartesian system with X, and Y axis, the Plane
coordinates are expressed in ordered pair format
representing distances from the origin.
Northing (Latitude) is the northerly or southerly distance
perpendicular to the X (Easting) axis.
Easting (Departure) Easterly or westerly distance
perpendicular to the Y (Northing) axis.
The State Plane Coordinates are a common example of
the plane coordinate system. However, local ground
based coordinate systems are commonly used by
surveyors to ensure the mitigation of linear distortion.
Plane Coordinate Systems
9/25/201813
Plān:
a flat surface on which a straight line joining any two
points on it would wholly lie.

9/25/201814
State Plane Coordinates
N:1697814.29 E: 3157197.06
Plane Round Surface Projected to
Flat Surface.
Coordinates ≠ Ground
Distances

9/25/201815
Projected (Plane) Coordinate Systems
P(N,E,El)
 Tertiary Positions.
 Ordered Pairs on a flat plane.
 Not Geodetic or Geocentric.
 Plane is attached to one or
more points of the ellipsoid.
 Positions “projected” onto a
flat surface from the
ellipsoidal surface.
 Distortion WILL occur.

9/25/201816
Types of Coordinate Systems
Geocentric
ECEF Positions
Geodetic
Latitude Longitude Positions
Plane
Projected Positions

and Their Coordinate Systems
9/25/201817
Modified State Plane
N:697814.29 E: 157197.06 CSF: 1.00026
Sometimes more modifications are made
Plane Truncate Coordinates &
Apply a Scale Factor.
Coordinates = Ground
Distances

and Their Coordinate Systems
9/25/201818
Earth Centered, Earth Fixed
P(x,y,z)
Geocentric GPS/GNSS Points
Traditional Cartesian
Latitude & Longitude
DD°MM’SS.SSSS”N, DDD°MMM’SS.SSSS”W
Geodetic Differs with Ellipsoid
GRS80 Vs. Clarks
State Plane Coordinates
N:1697814.29 E: 3157197.06
Plane Round Surface Projected to
Flat Surface.
Coordinates ≠ Ground
Distances
Modified State Plane
N:697814.29 E: 157197.06 CSF: 1.00026
Plane Truncate Coordinates &
Apply a Scale Factor.
Coordinates = Ground
Distances

9/25/201820
Neil Degrasse Tyson
Shape of the Earth
Neil Degrasse Tyson. Seth Rogan Podcast (2018)

9/25/201821
Kansas Really Is Flatter Than a Pancake
…Sort of
Mark Fonstad, William Pugatch and Brandon Vogt. Improbable Research (2003).

The Ellipsoid
9/25/201822
• Biaxial ellipsoid has dimensions a,
and b
• Mathematical approximation of the
shape and size of the earth.
• Defined by two dimensions; a Semi-
Major axis, (a), and a Semi-minor
axis, (b).
• The flattening ratio, (ƒ) of an
ellipsoid.
• ƒ = (a-b)/a
• The eccentricity, (e) is how much
the ellipsoid deviates from a circle.
• e = √(2 ƒ - ƒ 2).
• Typical notation: A precise
calculation of the ellipsoid can be
performed with the Semi-major
dimension, (a), and the flattening
ratio, ƒ.

The GEOID
9/25/201824
• Attaches to the X,Y,Z
Geocentric Reference
Frame.
• Model of the earths
gravitational surface.
• Used to obtain more
accurate elevations
• Geoid models must be
paired with the correct
horizontal datum.
Example: GEOID 12B should
be utilized in the NAD83
(2011) datum.

The GEOID and Ellipsoid
Deflection of the Vertical
9/25/201825
What is the LaPlace
Correction, anyway?
Ellipsoid
Geoid

Cylindrical Conical Planar
Types of Projections
9/25/201827

9/25/201828
Projection Orientations
Cylindrical
Conical
(or Cylindrical)

9/25/201829
SOME… Classical Projections

9/25/201830
SOME…. Polyhedral Projections

However….
 All projections will result in
some sort of distortion.
 Even a conformal projection
will distort angles over a
large enough area.
The Secant Lambert Conformal
Conic (SLCC) projection is one of
many types of projection
methodologies. This conformal
conic projection preserves the
integrity of angles. However linear
distortion will be present within the
design area of the projected
coordinate system.
Conformal Projection
9/25/201831
Conformal Projection:
In Cartography.
A map projection in which angles formed by lines are
preserved: a map made using this projection preserves
the shape of any small area.
Secant: Projection Plane Intersects the ellipsoidal surface in 2
locations, hence 2 standard parallels.
Tangent: Projection Plane intersects the ellipsoidal surface in
exactly 1 location, hence 1 standard parallel.
Non-intersecting: Projection Plane does not intersect the
ellipsoidal surface at all.

9/25/201832
The Conformal Lambert Conical
Secant Projection

9/25/201833
Secant Projection
This is our projection area
that will become our
coordinate system

9/25/201834
Secant Projection

Projections
Lambert Conformal Conic (SLCC) projection
9/25/201835
Standard Parallels are the only points (lines) in this projection
with a scale of 1. Standard parallels are important in a projection
because they essentially function as the boundaries.
The Origin is the anchor point for the projection. It is the point
from which Northing and Eastings are derived.
The Central Meridian. (in a coordinate systems) A line of
longitude that defines the center of a projected coordinate
system. In planar rectangular coordinate systems of limited
extent, such as state plane, grid north coincides with true north
at the central meridian.
The False Northing (map projections) The linear value added to
all y-coordinates of a map projection so that none of the values
in the geographic region being mapped are negative.
The False Easting (map projections) The linear value added to
all x-coordinates of a map projection so that none of the values
in the geographic region being mapped are negative.
The false northing and easting is often used to ensure that there are no negative coordinate
values within the projection.

Horizontal Datum
Orientation & Initial Point
9/25/201838
• Attaches the ellipsoid to a physical
point on the earth’s surface, known
as an initial point.
• Points chosen were best suited for
a particular region.
• Historically observatories were
used.
• Five Parameters:
1. Semi-Major (a)
2. Semi-Minor (b)
3. Latitude
4. Longitude
5. Azimuth to reference point
• Makes geodetic datums possible.
• NAD 1927 fixed at Meade’s Ranch
in Kansas and uses Clarke 1866
Spheroid.

Horizontal Datum
Geocentric Datums
9/25/201839
• Attaches the ellipsoid to a theoretical point at the
center of the earth, known as a geocentric
system.
• NAD 1983 used new technologies to take
measurements.
• Satellite Laser Ranging (SLR)
• Lunar Laser Ranging (LLR)
• Very Long Baseline Interferometry (VLBI)
• Doppler Orbitography (DORIS)
• Some (relatively few) GPS
• Realizations are needed to become useful in
Land Surveying.
• Realization can be thought of as a snapshot in
time. The center of the earth is a fixed point,
whereas the surface is constantly moving and
changing.
• Readjusted to incorporate additional “survey
measurements & observations”.
• Classical survey measurements
• EDM-measured baselines
• A lot of GPS observations
• Other
• NAD 1983 is fixed at, and attached to the center
of the earth, uses the GRS80 Spheroid, and
requires realizations.

Horizontal Datum
9/25/201840
A horizontal datum forms the
basis for computations of
horizontal positions which may be
defined at an origin point of an
ellipsoid such that the center of the
ellipsoid coincides with the earth’s
center, or actual fixed points on the
earth’s surface. A proper
horizontal datum will define:
• The dimensions of the
reference ellipsoid.
• The orientation of the
coordinate system.
• The location of the origin of the
coordinate system.
So really… A datum essentially
defines the coordinate system.
o NAD27
o NAD83
o NAD83 (2011)
o NAD83 (CORS96)
o 2022 Modernization

HEIGHTS (……AND GEOIDS)
9/25/201841

Heights
Ellipsoidal Height
9/25/201842
The Ellipsoidal Heights is the distance from the ellipsoid to a corresponding
point on the surface of the Earth. The height is measured along a line
perpendicular to the ellipsoid. This distance is known by more than one name. It
is called the ellipsoidal height and is also called the geodetic height, and it is
usually symbolized by h.
Note: Not all h values are the same, and the horizontal datum CAN impact h.

Heights
What is a Geoid and what is a Geoid Height
9/25/201843
The Geoid is an equipotential surface that best fits mean sea level. Meaning
that, across the geoid, the potential of gravity is always the same. The geoid and
mean sea level could be the same if the oceans of the world could be utterly still,
completely free of currents, tides, friction, variations in temperature, and all
other physical forces, except gravity.
A Geoid Height is the ellipsoidal height
from an ellipsoidal datum to a geoid.
That means that geoid height models are
directly tied to the geoid and ellipsoid that
define them. In other words, geoid height
models are not interchangeable.
Note that the geoid is a vertical datum surface.

Heights
Putting it together
9/25/201844
Orthometric Height (H) = h-N
Ellipsoidal Height (h) = H+N
Geoid Height (N) = h-H
Keep this in mind: An elevation that was established by differential leveling
should not be used to determine an Ellipsoidal Height because the level is
based upon gravity. However, this is negligible in most cases but reinforces the
need to ensure geoids are properly paired with the correct horizontal datum.

Scale Factor
9/25/201846
An Elevation Factor (EF) is determined by the ratio of the ground distance (DGND) to the corresponding geodetic
distance (DGDC): EF = DGND / DGDC.
The Scale Factor (SF) is determined by the ratio of the geodetic distance (DGDC) to the corresponding grid distance
(DGRD): SF = DGDC / DGRD.
A Combined Scale Factor (CSF) is perhaps the most useful, and is determined by the ratio of the ground distance
(DGND) and the grid distance (DGRD): CSF = DGND / DGRD.

Scale Factor
Grid to Ground and Ground to Grid
9/25/201847
Combined Scale Factor (CSF) = 1.00046
Grid to Ground
Ground Distance = Grid Distance x CSF
Ground to Grid
Grid Distance = Grid Distance/CSF
The horizontal distance between two section corners is measured on the surface
with a GPS unit, and has a ground distance of 5,280’. To convert the distance to
grid we take the ground distance (5280’) and divide it by the CSF (1.00046), which
gives us 5277.57’.
The horizontal distance between two section corners is measured on the surface
with a GPS unit, and has a grid distance of 5,280’. To convert the distance to grid
we take the ground distance (5280) and multiply it by the CSF (1.00046), which
gives us 5282.43’.

Modified Projection
9/25/201848
The Modified Projection is a method commonly used to
mitigate the adverse effects that elevation has on distances
such that ground distances are more accurately
represented. A correctly modified projection must be
defined in such a way that the coordinate values are easily
distinguishable between modified coordinate values and
non-modified coordinate values. This is typically
accomplished by a standard truncation of the Northing and
Easting coordinate values. The Modified System also
requires a combined scale factor, (CSF).
The Denver Water custom grids are a modification of the
Colorado State Plane coordinate system.

Modified Projection
9/25/201849
Denver Water Modified State Plane Coordinate Systems
State Plane to DW Grid DW Grid to State Plane
𝑆𝑆𝑆𝑆𝑁𝑁 − 𝑇𝑇𝑇𝑇𝑁𝑁 ∗ 𝐶𝐶𝐶𝐶𝐶𝐶 = 𝐷𝐷𝐷𝐷𝑁𝑁 (𝐷𝐷𝐷𝐷𝑁𝑁/𝐶𝐶𝐶𝐶𝐶𝐶) + 𝑇𝑇𝑇𝑇𝑁𝑁 = 𝑆𝑆𝑆𝑆𝑁𝑁
𝑆𝑆𝑆𝑆𝐸𝐸 − 𝑇𝑇𝑇𝑇𝐸𝐸 ∗ 𝐶𝐶𝐶𝐶 = 𝐷𝐷𝐷𝐷𝐸𝐸 (𝐷𝐷𝐷𝐷𝐸𝐸/𝐶𝐶𝐶𝐶𝐶𝐶) + 𝑇𝑇𝑇𝑇𝐸𝐸 = 𝑆𝑆𝑆𝑆𝐸𝐸
CSF: Combined Scale Factor
DW: Denver Water Coordinate
N=Northing E=Easting
SP: State Plane Coordinate
Tr: Coordinate Truncation

Create a Ground Based
Projection for your GIS Project…
9/25/201850
1. Work with your surveyor
2. Request the projection definition from the surveyor
3. Find a Combined Scale Factor
4. Create a Projection
5. Modify and rename
6. Test the Projection
7. Create a prj to utilize in the future or share data

Request The Projection
Definition From The Surveyor
9/25/201851
Important Parameters to
Request
 Units
 Projection Type
 False N and E
 Central Meridian
 Scale Factor
 Latitude of Origin
 Geographic coordinate
System (Geodetic
Coordinate System)
Denver Water Projection Definition

How to Determine a Combined
Scale Factor
9/25/201852
Scale Factor Resources
 Surveyors
 NGS Datasheets published
by the National Geodetic
Survey – NGS
https://www.ngs.noaa.gov/
datasheets/
 Published Data on ArcGIS
Online – All Portal (Just be
mindful of the person or
organization publishing the
data)
 https://www.ngs.noaa.gov/datasheets/
A Shapefile from NGS – You
Must Keep It Up To Date.
There is no data service for
this.

Scale Factor
9/25/201853
NGS Data Explorer
 Launch Map
 Zoom to Project Area
 Map Layers
 Select the Mark Types of
Interest and “Find Marks”
 Pay Close Attention to
“Control Types” to make
sure you utilize good data.

Scale Factor
9/25/201854
Find the Datasheet
 Examine the Basic
Information to
determine if it is what
you are looking for.
 Launch “Datasheet”

Scale Factor
9/25/201855
https://www.ngs.noaa.gov/datasheets/
Combined Scale Factor
0.99993765 sFT
Colorado Central SP
Select the appropriate scale factor.
In this case Colorado State Plane
Central Zone in Survey Feet.

Scale Factor
9/25/201856
https://www.ngs.noaa.gov/datasheets/
Select Several Horizontal
Controls around the
Project Area and average the CSF
Calculate Average
0.99993765
0.99993614
0.99993739
Avg: 0.99993706
1/x: 1.00006294
Note:
1/x is the inverse.
1/0.99993706 = 1.00006294
Most software asks for the
inverse input.

How to Build a Ground Based
Projection
9/25/201857
“Copy and Modify” the
projected Coordinate System
that you want to start with.
In this case we are going to
modify Colorado Central Zone
State Plane.

How to Build a Ground Based
Projection
9/25/201858
Entering Parameters
Change the name of the
projection to something
meaningful.
Enter the previously
computed scale factor.
Don’t forget to add to your
favorites.
Note:
By utilizing this method, the
coordinate values will look
very similar to plain State
plane coordinates.

Examine Grid V. Ground
9/25/201859
Grid v. Ground
Set the map Coordinate
System to the modified
projection.
Import field measured points.
At this point, you should have
“ground” information
resulting in ground distances.

Examine Ground V. Grid
9/25/201860
Ground v. Grid
System back to State Plane.
Leave the field measured
point feature in the map.
resulting in grid distances.
From the north end of the
dam to the south end of the
dam:
Modified Projection = 4111.53
SP Projection = 4111.27

Examine Grid V. Ground
9/25/201861
Grid v. Ground
System to the modified
projection.
Import field measured points.
resulting in ground distances.

Examine Ground V. Grid
9/25/201862
Ground v. Grid
System back to State Plane.
Leave the field measured
point feature in the map.
resulting in grid distances.
From the north end of the
dam to the south end of the
dam:
Mod. Projection = 7,834,490
SP Projection = 7,833,504

Future of the NSRS
9/25/201863
 Modernization of the NSRS – 2022
- Horizontal Displacement of apx. 3.6 ft.
- Vertical Displacement of apx. 2.9 ft.
 SPCS 2022
- NGS currently designing
- Potential for LDP designs
 Contemporary
- Future could see time dependent positioning
- Largely driven by technology
NATRF 2022 Approximate Horizontal Displacement NATRF 2022 Approximate Vertical Displacement

Good Geodetic Resources
9/25/201864
National Geodetic Survey
https://geodesy.noaa.gov/#
Michael Dennis – Geodetic Analysis
https://geodeticanalysis.com/resources/
Dave Doyle – Base9 Geodetic Consulting Services
https://www.base9geodesy.com/publications
Jan Van Sickle – Basic GIS Coordinates
Available on Amazon

B. John Hunter, PLS
Geodetic Surveyor
Denver Water
Colorado NGS Geomatics Coordinator
CoCoordinator@plsc.net
john.hunter@denverwater.org
720.883.6508

2018 GIS in the Rockies Workshop: Coordinate Systems and Projections

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