An Illustrated Guide to Geodesy

for Engineering,
Surveying,
and GIS
Michael Dennis, RLS, PE
michael.dennis@noaa.gov

Esri AEC Summit

Manchester Grand Hyatt
July 12-15, 2014 • San Diego, CA

®
Why should we care about geodesy?

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 The Plan
• Introducing the National Geodetic Survey
• GIS and the National Spatial Reference System (NSRS)
• Geodetic concepts – illustrated!
• Focus today is on geometric (“horizontal”) datums
– Connecting data to the Earth: Datums
– A complicated topic: Datum transformations
– Nothing is constant except change: Time dependence
– Specific example: “WGS 84” ßà “NAD 83”
• Other topics of Great Interest (pursue as time allows)
– How data are displayed and analyzed: Map projections
– How high? How deep? Where will water go? Gravity and heights
– How good is it? How do you know? Accuracy (vs. precision)
– Putting it all together: Metadata
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 NGS: Who we are is where you are
• Been around a long time (since 1807)
– Became National Geodetic Survey in 1970 (when NOAA created)
• Keepers of the National Spatial Reference System (NSRS)
– Define, maintain, provide access for US and territories
– Position, height, scale, gravity, orientation
– …and how they change with time
• Products & Services
– Geodetic control (active and passive)
– Data, models, and imagery (gravity, geoid, aerial imagery)
– Tools and services (online and desktop software and services)
– Standards, specifications, guidelines, and education

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The NSRS is the foundation for GIS

Land Ownership

Transportation

Surface Waters

Boundaries
Elevation
Aerial Imagery

NSRS Control
Geodetic
 What is a (geometric) “datum”?
• Geometric (“horizontal”) datums
– a.k.a. “geographic coordinate systems”
– Basis for determining positions on the Earth
– Modern ones are 4-D (time is 4th dimension)
• Lat, lon, height (or Earth-Centered, Earth-Fixed XYZ) + velocities
– Ellipsoid (“spheroid”) by itself is NOT a datum
• ~Same ellipsoid for NAD 83 and WGS 84, but differ by ~2 m
• Includes “local” and “global” datums
– “Local” (regional) datums (e.g., NAD 83)
– “Global” datums (e.g., WGS 84)
• Vertical datums another topic for another time…
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 “The Figure of the Earth”
Best-fit spherical Too big by 9 miles
Earth model at the poles

Point #1
San Diego
Too small by Earth mass center Too small by
4 miles at 4 miles at
the equator the equator
Mean
Earth radius,
R ≈ 3959 miles
Equatorial
plane

Geoid
Too big by 9 miles (“mean sea level”)
at the poles
 Earth model: Ellipsoid of revolution
Best-fit ellipsoid a = 6,372,137.000 m ≈ 3963 mi
(e.g., GRS-80, WGS-84) b = 6,356,752.314 m ≈ 3950 mi
Ellipsoid flattening
f = (a – b)/a ≈ 0.335%
1/f ≈ 298.25722
b = semi-minor axis
Point #1 (polar radius)
San Diego
Earth mass center

a = semi-major axis
(radius of equatorial plane)
Equatorial
plane

Ellipsoid fits geoid to Geoid

within about ±100 m worldwide (“mean sea level”)
 Geospatial Codependence
the first step is admitting you have a problem
• Datums must be “realized”
– Connected to Earth by observations (e.g., GNSS)
– New realizations improve accuracy of coordinates
– A single datum can have multiple realizations
• Interrelationships given by “datum transformations”
– Mathematical methods for converting between datums
– Needed if combining data based on different datums
– There are many different kinds and they can vary greatly
• Modern datum definitions are accurate but complex
– Will focus on two here: NAD 83 and WGS 84.
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A (very) brief history of NAD 83 datum
• Original realization completed in 1986
– Almost entirely classical (optical) observations
• “High Accuracy Reference Network” (HARN)
realizations (1990s)
– Done essentially state-by-state
– Based on GNSS but classical stations included
• National Re-Adjustment of 2007
– NAD 83(NSRS2007/CORS96) epoch 2002.00
– Nationwide adjustment (GNSS only)
• NAD 83 (2011/PA11/MA11) epoch 2010.00
– Also nationwide GNSS-only adjustment
– This is NOT a new datum! (still NAD 83)

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 Alaska

CONUS
Pacific
(MA11)

Pacific (PA11)

GIS is only for low spatial accuracy?

Median accuracy (95% conf.):

0.9 cm horiz, 1.5 cm height
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A (very) brief history of WGS 84 datum
• Original realization completed in 1987
– “Same” as original NAD 83 (to within ±1-2 m)
• WGS 84 (G730) — adopted Jan 1994
– Aligned with ITRF91
• WGS 84 (G873) — adopted Sep 1996
– Aligned with ITRF94
• WGS 84 (G1150) — adopted Jan 2002
– Aligned with ITRF2000 (at epoch 2001.00)
• WGS 84 (G1674) — adopted Feb 2012
– Aligned with ITRF2008 (at epoch 2005.00)
• WGS 84 (G1762) — adopted Oct 2013
– Also aligned with ITRF2008 (at epoch 2005.00)
– Note that current NAD 83 is epoch 2010.00
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Understanding geodetic coordinates
• Positions for entire Earth (or large part of Earth)
– For modern datums these are 3-D
• With velocities they are 4-D
– Here concerned with geometric coordinates
• Two main types:
– Latitude, longitude, and ellipsoid height: φ, λ, h
– Earth-Centered, Earth Fixed (ECEF) Cartesian: X, Y, Z
• Used for many types of geodetic computations
– Can covert between both types without error

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 Earth-Centered Earth-Fixed (ECEF) coordinates
Ellipsoid
(e.g., GRS-80, WGS-84) +Z axis (parallel to axis of rotation)

+Y axis (90°E)
Point #1, San Diego
+Z1
Coordinates:
(–X1, –Y1, +Z1)
(φ1, λ1, h1)
h 1
Earth mass center +X axis
–X axis –X1
(180° φ 1
(Prime
W) meridian)
–Y1 λ 1

Equatorial
plane

Geoid
–Y axis (90°W) (“mean sea level”)
–Z axis
 Earth-Centered Earth-Fixed (ECEF) coordinates
Ellipsoid
(e.g., GRS-80, WGS-84) +Z axis (parallel to axis of rotation)

+Y axis (90°E)
Point #1, San Diego
+Z1
Coordinates:
(–X1, –Y1, +Z1)
(φ1, λ1, h1)
h 1
Earth mass center +X axis
–X axis –X1 Where is San Diego Conference Center?
(180° φ (Prime
1
X = -2,451,510 m meridian)
W)
–Y λ1
Y1= -4,780,100 m
Equatorial Z = +3,426,640 m
plane
…is the same as:
Latitude, φ = 32°42’ 25” N
Longitude, λ = 117°09’ 05” W
Geoid
–Y axis (90°W) Ellipsoid height, h = -30 m sea level”)
(“mean
–Z axis
 Datum transformations
• Typical datum transformations
– 3-parameter: 3-dimensional translation of origin as ΔX, ΔY, ΔZ
– 7-parameter: 3 translations plus 3 rotations (one about each of
the axes) plus a scale
– 14-parameter: A 7-parameter where each parameter changes
with time (each has a velocity)
– Transformations that model tectonic displacement and other
distortion (e.g., NGS models in HTDP, GEOCON, and NADCON)
• Vertical datum transformations
– Can be simple shift or complex operation that models distortion
(e.g., GEOCON, VERTCON)

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Datum transformations

b2 b1

a1
a2

3-parameter
datum transformation
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Datum transformations
If datum changes
with time, each rotZ
component has a b1
velocity…

a1
b2

rotX
a2

3-parameter
7-parameter
14-parameter
rotY datum transformation
scale
 What to do…?

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 Based on 7 constant
transformation parameters
published by NGS (7 time-
dependent parameters ignored),
at input time = output time of
1997.0 (so tectonic velocities
irrelevant).

This is how the transformation

is implemented in most
commercial geospatial software.

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 Based on 14 transformation
parameters published by NGS at
input time = output time of 2010.0
(so tectonic velocities irrelevant)

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 Based on 14 transformation
parameters published by NGS at
input time = output time of 2005.0
(so tectonic velocities irrelevant)

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 Based on 14 transformation
parameters published by NGS
at input time of 2005.0 ≠
output time of 2010.0 (5 years
of tectonic movement).

This much more complex case

is the “correct” transformation.

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This 7-parameter transformation is equivalent to the
following commercial vendor transformations:
n ESRI: “WGS_1984_(ITRF08)_To_NAD_1983_2011” (108363)
• NOT “WGS_1984_(ITRF00)_To_NAD_1983_2011”, or the ~25+ other
NAD 83ßàWGS 84 transformations in ArcGIS 10.x
n Trimble: “ITRF to NAD 1983 (2011)”
• NOT “NAD 1983 (Conus)”, a “zero” transformation (used automatically
with State Plane)
n Topcon: “NAD83”
• NOT “NAD83_NO_TRANS”, another “zero” transformation

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 Where do you get “WGS 84”
coordinates?
• Autonomous positions (~3 – 5 m accuracy)
• Satellite Based Augmentation Systems (SBAS)
– WAAS solutions (1 – 2 m accuracy)
– OminStar HP (10 cm)
• Various GNSS positioning services (~1 cm)
– OPUS (on right side of solution report)
– AUSPOS
– CSRS-PPP
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How to check transformation: HTDP

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 Does this stuff really matter?
• Significant for accuracies better than ~1-2 m
– Can be problem for combining accurate datasets
– Requires understanding of modern datums
– Be careful when using “WGS 84”
• Which realization? At what epoch? At what level of accuracy?
• Things are moving, and it can make a difference
– e.g., San Diego moving 4.0 cm/yr NW w.r.t. Phoenix, AZ
• GNSS becoming more precise
– Autonomous positions soon better than 1-2 m
– But accuracy another issue… with respect to what?
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 New Datums for the U.S.
• Planned release in 2022
– Geometric datum: Aligned with ITRF/WGS 84
– Vertical datum: Based on gravimetric geoid
• How much will NSRS coordinates change?
– North America plate (CONUS and AK): Approx 0.8 to 1.6 m
– Pacific plate: Approx 3.4 (Midway) to 4.3 m (American Samoa)
– Mariana plate: Approx 1.1 to 1.4 m
• How much will NSRS ellipsoid height change?
– Approx -1.9 m (Puerto Rico) to +2.0 m (Guam)
• How much will NSRS CONUS orthometric height change?
– Approx +0.1 m (Florida) to -1.3 m (Washington)

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 NAD 83(2011) to
IGS08 at epoch 2022.0
 NAD 83(PA11) to
IGS08 at epoch 2022.0
 NAD 83(2011) to
IGS08 at epoch 2022.0
 Differential Tectonic Displacement

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 NAD 83(2011) coordinate shifts

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 Grid-based transformations

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 Different Flavors of NAD 83

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 GEOCON and GEOCON11
• Transformation grids for latest realizations of NAD 83
– GEOCON: NAD 83 “HARN” ßà NAD 83(NSRS2007/CORS96)
– GEOCON11: NAD 83(NSRS2007/CORS96) ßà NAD 83(2011)
– Both horizontal AND “vertical” (i.e., ellipsoid height)
– 1 arc-minute spatial resolution
– Include output that indicates “quality” of transformation
• Quantified using station within grid cell that is worst match with model
• Complication: Some states have > 1 HARN realization
• Current release only uses Bluebook format as input/output
– Difficult to use because format not readily available
– Will provide more flexible input and output soon

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GEOCON and GEOCON11 (beta)
GEOCON and GEOCON11 (beta)
 GEOCON and GEOCON11 (beta)

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GEOCON and GEOCON11 (beta)
GEOCON and GEOCON11 (beta)
GEOCON and GEOCON11 (beta)
 GEOCON and GEOCON11 (beta)

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GEOCON and GEOCON11 (beta)
 Map Projections
• GPS yields geodetic coordinates
– Usually given as latitude, longitude, and ellipsoid height
• Usually must be “projected” to be useable
– “Map projection” used to convert to “grid” coordinates
• i.e., Northing, Easting, “Elevation” (y, x, “z”)
– Essential for printed maps, computer display, CAD
– Projected coordinates are always distorted
• There is no way around this — it is a Fact of Life
• Cannot be eliminated — it can only be reduced
• Effect is computable — it is NOT the same as “error

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State Plane (AZ zone C) “linear distortion” (feet/mile)

+3.0 ft/mile

-3.0 ft/mile
State Plane (AZ zone C) “linear distortion” (feet/mile)

+3.0 ft/mile

-3.0 ft/mile
 State Plane (AZ zone C) “linear distortion” (feet/mile)

Central
meridian

Zero Zero
distortion distortion

W 1/6 2/3 1/6 E

Zone width to balance positive and negative distortion

(with respect to the ellipsoid)
Gravity, Heights, and Vertical Datums
• How high is it?
• How deep is it?
• Where will water go?
• What kind of heights (“elevations”) to use?
– Orthometric?
– Dynamic?
– Ellipsoid?

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 The ellipsoid, the geoid, and you
Deflection of the vertical

You are here

Earth
surface
Ellipsoid height, h
Orthometric height, H
Mean
sea level
Geoid height, NG

h≈
= H + NG
Note: Geoid height is negative everywhere in the coterminous US
(but it is positive in most of Alaska)
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Experimental
NGS geoid
models
(beta)
• xGEOID14A
and
xGEOID14B
• xGEOID14B
includes aerial
GRAV-D data
• Covers nearly one
quarter of Earth

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Orthometric height
change (meters)

NAVD 88 to new vertical datum

Estimated as NAVD 88 "zero" (datum)
surface minus NGS gravimetric geoid
 Spatial Accuracy (vs. Precision)
• How good is your data?
• How do you know?

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 NAD 83(2011) epoch 2010.00
Horizontal accuracy, cm (95% confidence)

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 NAD 83(2011) epoch 2010.00
Vertical accuracy, cm (95% confidence)

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 Some “accuracy” results to consider
(all from same set of actual GPS data)
• Horizontal “accuracy” computed as:
– ±0.6 feet
– ±0.8 feet
“Torture numbers, and they'll confess
– ±1.4 feet to anything.” ~Gregg Easterbrook
– ±1.8 feet
– ±2.7 feet “Statistics are like bikinis. What they
– ±3.8 feet reveal is suggestive, but what they
– ±6.5 feet conceal is vital.” ~Aaron Levenstein

– ±9.4 feet
• Is it “accuracy” or is it “precision”?
• What is the “confidence level”?
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 Scatter plot with respect to known coordinates
8.0

Sample size
n = 130

4.0
Delta north (ft)

0.0

-4.0

-8.0
-8.0 -4.0 0.0 4.0 8.0
Delta east (ft)
Scatter plot showing standard precision (39% confidence)
8.0

Circular
precision =
4.0 0.6 ft
Delta north (ft)

0.0

Precision ellipse
a = 0.8 ft
-4.0

-8.0
-8.0 -4.0 0.0 4.0 8.0
Delta east (ft)
Scatter plot showing scaled precision (95% confidence)
8.0

Circular
precision =
4.0 1.4 ft
Delta north (ft)

0.0

Precision ellipse
a = 1.8 ft
-4.0

-8.0
-8.0 -4.0 0.0 4.0 8.0
Delta east (ft)
Scatter plot showing standard accuracy (39% confidence)
8.0

Circular
4.0
accuracy = 2.7 ft
Delta north (ft)

0.0

-4.0
Accuracy ellipse
a = 3.8 ft

-8.0
-8.0 -4.0 0.0 4.0 8.0
Delta east (ft)
Scatter plot showing scaled accuracy (95% confidence)
8.0
Circular
accuracy =
6.5 ft (NSSDA)

4.0
Delta north (ft)

0.0

-4.0

Accuracy ellipse
a = 9.4 ft
-8.0
-8.0 -4.0 0.0 4.0 8.0
Delta east (ft)
 Some “accuracy” results to consider
(all from same set of actual GPS data)
• Horizontal “accuracy” computed as:
– ±0.6 feet Accuracy
(precision per
circleNMAS
at 39%(circular) = 5.7 ft
confidence)
(90% confidence)
– ±0.8 feet (precision ellipse at 39% confidence)
Accuracy RMSE (radial) = 3.9 ft
– ±1.4 feet (precision circle at 95% confidence)
(~66% confidence)
– ±1.8 feet (precision ellipse at 95% confidence)
Razzle-dazzle accuracy = 0.4 ft
– ±2.7 feet (accuracy
(1% circle at 39% confidence)
confidence)
– ±3.8 feet (accuracy ellipse at 39% confidence)
– ±6.5 feet per NSSDA (95% confidence)
– ±9.4 feet (accuracy ellipse at 95% confidence)
• Is it “accuracy” or is it “precision”?
• What is the “confidence level”?
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 Accuracy scatter plot (95% confidence)
8.0
Circular
accuracy =
6.5 ft (NSSDA)

4.0
Delta north (ft)

0.0

-4.0

Accuracy ellipse
a = 9.4 ft
-8.0
-8.0 -4.0 0.0 4.0 8.0
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Delta east (ft)
Transformed accuracy scatter plot (95% confidence)
8.0
Circular
accuracy =
1.4 ft (NSSDA)

4.0
Delta north (ft)

Applied published
“CORS96”datum
0.0 transformation
(ITRF 00 to NAD 83
at time 1997.0)

-4.0

Accuracy ellipse
a = 1.8 ft
-8.0
-8.0 -4.0 0.0 4.0 8.0
Delta east (ft)
 Spatial Data Requirements
• Completely define the coordinate system
– Linear unit (meter, foot, what kind of foot?)
– Geodetic and vertical datums
– Map projection parameters
• Is “distortion” an issue for your project . . . ?
• Require ties to published control
– Best source for control: National Geodetic Survey (NGS)
• But control must be more accurate than surveying or mapping
method used

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 Spatial Data Requirements
• Specify accuracy requirements
– Use objective, robust, defensible methods
• Make use of published standards (e.g., NSSDA)
• Least-squares network adjustment (but no cheating!)
– Remember: Accuracy may not be same as precision
• Documentation is essential (metadata!)
– Report methods, procedures, and results for creating final
deliverables
– Must include any coordinate transformations
• E.g., datum transformations, “rubber sheeting”

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 Documentation (metadata!)
• Essential for correctly georeferencing data
– Especially important if a “custom” coordinate system is used
– Should rarely have to resort to approximate methods (e.g., “best-fit”,
“rubber sheeting”)
• Metadata needs to be AVAILABLE and kept with the data!
– Best is to embed it with the data so that it cannot be lost or easily
ignored
– Also helpful if it is CORRECT
• Consistent with spirit of “leaving footsteps…”
– Others should be able to retrace the original surveying or mapping
product (without having to contact the person who created it!)

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Example of typical geospatial metadata . . .

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 Conclusions
• Geodetic knowledge needed for correct georeferencing
– Becomes more important as spatial accuracy increases
– Driven by high precision and low cost of GNSS (a geodetic tool)
• High-accuracy geodetic transformations are complicated
– Time-dependence especially complex for differential tectonic motion
– Datasets representing different times difficult to spatially align
• Metadata (documentation) is essential
– Improves reliability and accuracy of data
– Increases value and usefulness of spatial data
– Needed all geospatial data (GIS, surveying, engineering, etc.)

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 More information…
geodesy.noaa.gov

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 An Illustrated Guide to Geodesy
for Engineering,
Surveying,
and GIS
Questions?

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