The Iowa Regional Coordinate SystemPutting “Grid”

at “Ground”

Michael L. Dennis, RLS, PEGeodetic Analysis, LLC

Oregon State UniversityASCE Land Survey ConferenceScheman Building, Ames, IowaOctober 19, 2016 © 2016 Michael L. Dennis

The problem Map distance ≠ “ground” distance

• Due to map projection linear distortion

• Often called “grid vs. ground” problem

A problem for some geospatial products

• Engineering & construction plans

• Survey plats and legal descriptions

• As-built surveys and facilities management

Just click this button in your favorite geospatial software!

GroundCoordinates

GroundCoordinates

What is the solution?

Michael L. Dennis, RLS, PE

mld@geodetic.xyz

Presentation available at www.geodetic.xyz

Questions?

…and how we will get there

Map projection distortion — what is it?

Methods for reducing projection linear distortion

Low distortion projections (LDPs)

The Iowa Regional Coordinate System (IaRCS)

Spatial data documentation (metadata!)

Using the IaRCS in software

Distortion calculations

Most important: Ask questions! Make comments!

Where we are going…

Iowa Regional Coordinate System

Let’s talk about State Plane When was it created?

Where was it first used?

Why was it created?

Why and how is it used today?

Disadvantages

• Distortion often too great for “ground” distances

• Often “modified” to get “ground” coordinates

• No standardized way to “modify”

• “Modification” will generally NOT minimize distortion

OBJECTIVE: To contemplate coordinate systems!

http://www.geodesy.noaa.gov/datums/newdatums/nad-83_state-plane-leg-coord_flyer.pdf8

What is solution to distortion “problem”?

Make map distance = “ground” distance?

• This is IMPOSSIBLE

• No such thing as “ground” coordinates

• However, linear distortion CAN be MINIMIZED

“Low distortion projections” (LDPs)

• Based on existing CONFORMAL projection types

• Linear distortion at point same in all directions

• Only a few appropriate existing projections available

Why bother with LDPs? Rigorously defined and widely supported

Satisfies needs of surveying, engineering, and GIS

Enables direct use of survey data in a GIS

Reduces proliferation of local systems

Facilitates data transferability

Optimally minimize distortion over area of interest

If designing a custom coordinate system, do it right the first time maximize benefit

Gnomonic projection

(non-conformal)

Stereographic projection

(conformal)

Orthographic projection

(non-conformal)

Ray source at infinity

Ray source at opposite side of Earth

Ray source at center of Earth

Projection surface Projection surface Projection surface

Can think of projection as “light rays” projecting onto surface

However, this only works for a few sphere-based projections

e.g., Lambert conformal conic

Polar aspect

Oblique aspect

Central parallel

Standard parallels

Origin

e.g., stereographic

One-parallel Two-parallel

Conic projectionsPlanar (azimuthal) projections

Developable surfaces: Planes and cones

Examples given are CONFORMAL projections

Cylindrical projections

Central meridian Skew

axis

Equator

“Regular” aspect Transverse

aspectOblique aspect

e.g., Mercator e.g., transverse Mercator

e.g., oblique Mercator

Developable surfaces: Cylinders

Examples given are CONFORMAL projections

ConformalityWhat it is and why it matters

Conformality

• Scale (distortion) same in every direction at a point

• Angles on Earth same as on map

A projection is a compromise

• Some characteristic must be “sacrificed”

• e.g., no projection can be both conformal and equal area

Only conformal projections make sense for LDPs

Map Projection Distortion

5279.5 ft “grid”

Alaska State Plane of 1983, Zone 4 (5004)

5280.0 ft “ground”52

79.5

ft “

grid

”52

80.0

ft “

grou

nd”

Δ = −0.5 ft

Δ = −0.5 ft

Transverse Mercator

projection (conformal)

639.88 acΔ = −0.12 ac

Anchorage(assume h ≈ 0)

Map Projection Distortion

5260.5 ft “grid”

Alaska Albers Equal Area Projection

Anchorage(assume h ≈ 0)

5280.0 ft “ground”52

99.6

ft “

grid

”52

80.0

ft “

grou

nd”

Δ = −19.5 ft

Δ = +19.6 ft

Distortion (scale)

varies with direction for non-

conformal projections

!640.00 ac

Δ = 0.00 ac

Alaska Albers Equal Area

Lambert Conformal Conic for AK

Transverse Mercator for AK

Types of Map Projection Distortion

Angular distortion. Equals convergence (mapping) angle

for conformal projections (e.g., Transverse Mercator)

Linear distortion. Difference in distance between a pair of

grid (map) coordinates when compared to the “true”

horizontal (“ground”) distance

• Can express as ratio of distorted length to “true” length

E.g., feet of distortion per mile, or as mm/km (= ppm)

• Distortion can be negative or positive

NEGATIVE: Grid distance less than true distance

POSITIVE: Grid distance greater than true distance

Zone width to balance positive and negative scale error with respect to ellipsoid

Projection of ellipsoid on developable surface

14.5% 71% 14.5%

Projectionaxis

Secant developable surface

Ellipsoid surface

Topographic surface

Tangent developable surface

Non-intersecting developable surface k0 > 1

k0 = 1

k0 < 1

k > 1 everywhere

k = 1 at axis, k > elsewhere

Secant, tangent, and non-intersecting developable surfaces

Grid distance < ellipsoidal distance

(distortion < 0)

Grid distance > ellipsoidal distance

(distortion > 0)

Projectionaxis

Projection surface (secant)

Ellipsoid surface

Ellipsoid distance

Topographic surface

Horizontal ground

distance

Ellipsoid distance

> ellipsoid distanceand

> grid distance

Linear distortion with respect to ellipsoid and ground

Maximum projection

zone width

Maximum linear horizontal distortion

Parts per

millionFeet per mile

Ratio

(absolute value)

16 miles ±1 ppm ±0.005 ft/mi 1 : 1,000,000

50 miles ±10 ppm ±0.053 ft/mi 1 : 100,000

71 miles ±20 ppm ±0.11 ft/mi 1 : 50,000

Horizontal distortion due to Earth curvature

Maximum projection

zone width

Maximum linear horizontal distortion

Parts per

millionFeet per mile

Ratio

(absolute value)

16 miles ±1 ppm ±0.005 ft/mi 1 : 1,000,000

50 miles ±10 ppm ±0.053 ft/mi 1 : 100,000

71 miles ±20 ppm ±0.11 ft/mi 1 : 50,000

112 miles ±50 ppm ±0.26 ft/mi 1 : 20,000

158 miles* ±100 ppm ±0.53 ft/mi 1 : 10,000

317 miles** ±400 ppm ±2.11 ft/mi 1 : 2,500

Horizontal distortion due to Earth curvature

Height below (–) and above (+)

projection surface

Maximum linear horizontal distortion

Parts per

million

Feet per mile

Ratio

(absolute value)

±100 ft ±4.8 ppm ±0.025 ft/mi ~1 : 209,000

±400 ft ±19 ppm ±0.10 ft/mi ~1 : 52,000

±1000 ft ±48 ppm ±0.25 ft/mi ~1 : 21,000

Horizontal distortion due to height above ellipsoid

Height below (–) and above (+)

projection surface

Maximum linear horizontal distortion

Parts per

million

Feet per mile

Ratio

(absolute value)

±100 ft ±4.8 ppm ±0.025 ft/mi ~1 : 209,000

±400 ft ±19 ppm ±0.10 ft/mi ~1 : 52,000

±1000 ft ±48 ppm ±0.25 ft/mi ~1 : 21,000

+2000 ft –96 ppm –0.51 ft/mi ~1 : 10,500

+3300 ft* –158 ppm –0.83 ft/mi ~1 : 6300

+14,400 ft** –690 ppm –3.6 ft/mi ~1 : 1450

Horizontal distortion due to height above ellipsoid

Minimizing linear distortion

Methods for reducing linear distortion:

• Scale existing projection (e.g., State Plane) “to ground”

• Scale ellipsoid “to ground” and use for custom projection

• Design custom projection “at ground”

All do same thing:

• Projection developable surface “at” topographic surface

But not all perform the same:

• Certain methods better at minimizing distortion

Design location

State Plane –Lambert Conformal Conic projectionSecant

Topographic surfaceStandard parallel

Central parallel

Standard parallel

Ellipsoid

State Plane –Lambert Conformal Conic projectionScaled to “ground”

Design location

Topographic surface

Central parallel

Ellipsoid

Design location

Topographic surface

Low Distortion Projection –Lambert Conformal Conic projectionBest-fit to topographic surface

Central parallel

Ellipsoid

General LDP design approach for sloping topography

Choose existing projection type

• Slope east-west Transverse Mercator (TM)

• Slope north-south Lambert Conformal Conic (LCC)

• Slope oblique direction Oblique Mercator (OM)(or Oblique Stereographic, but not recommended)

“Flat” areas with long dimension > ~30 miles (~50 km)

• Choose projection based on long dimension

• TM if long N-S, LCC if long E-W, OM if long oblique direction

• But offsetting projection axis can further reduce distortion

Choose several/many design points

• Distributed throughout project area

• Projection axis through centroid

• Scale projection using mean ellipsoid height

• These steps are to START the process…

General LDP design approach for sloping topography

Change projection axis location/orientation

• To minimize distortion variation

• e.g., range and standard deviation

Change projection scale

• Distortion changes by same amount everywhere

Make average distortion = zero

OR balance positive and negative distortion

General LDP design approach for sloping topography

Be aware:

Designing an LDP is an OPTIMIZATION problem

Often want LEAST distortion over LARGEST area

• These goals are at odds with one another

• So design may not be a simple process

General LDP design approach for sloping topography

Topography slopes down ~0.6% (~0.3°) at azimuth of 17°

The ellipsoid, the geoid, and you

Earth surface

Orthometric height, H

Geoid height, NG

Ellipsoid height, h

Deflection of the vertical

Mean sea level

h ≈ H + NG

Note: Geoid height is negative everywhere in the coterminous US

h = H + NG

You are here

Mean = +9 ppmRange = 68 ppmStdev = ±25 ppm

Mean = +2 ppmRange = 41 ppmStdev = ±15 ppm

Mean = -3 ppmRange = 33 ppmStdev = ±11 ppm

Mean = +4 ppmRange = 24 ppmStdev = ±8 ppm

Mean = +3 ppmRange = 19 ppmStdev = ±8 ppm