week 5-fnr

Where are we? Where are we
going? Are we there yet?

Early solutions:
marking trails with piles of stones (kairns)
memory aids
navigating by stars (requires clear nights and careful
measurements)
most widely used for centuries ... locations accurate
to within a mile or so

More recent solutions

Lighthouses
Coastal radio-based navigational beacons (limited outside of coastal areas)
Sat-Nav: low orbit satellites ...problems with small movements of receivers

The resulting end-product is what

we now call Global
Positioning System (GPS)

question

system (constellation) of ~20-30 satellites in high
altitude orbits (cost ~ $15 billion)
Coded satellite signals that can be processed in a GPS
receiver to compute position, velocity, and time
First one launched in 1978.
As of Jan. 31, 2023, there are 31 operational GPS sats.

US System is called Navstar

Other countries have their own GPS constellations

CChina: BeIDou Navigation Satellite System
(BDS)
EU: Galileo
Russia: Global Navigation Satellite System
(GLONASS)
India: Indian Regional Navigation Satellite
System (IRNSS) / Navigation Indian
Constellation (NavIC)
Japan: Quasi-Zenith Satellite System (QZSS)

What does “position” mean?

GPS is used to measure position on the surface of the
Earth, consisting of:
Latitude and Longitude

Altitude (height)
Velocity (movement)

GPS components

System includes 3 parts:
1. Space (GPS satellite vehicles, or SVs)
2. Control (ground tracking stations)
3. Users (GPS reciever)

Satellites

Fleet of radio-emitting and –receiving satellites
Operated by US Dept. of Defense
“...the U.S. Air Force's 2nd Space Operations Squadron
(2SOPS) and the Air Force Reserve's 19th Space Operations
Squadron (19SOPS) at Schriever Air Force Base, Colorado.
Together, 2SOPS and 19SOPS — nicknamed Team Blackjack
— keep the GPS satellites flying on a 24/7 basis, with
continuous availability and high accuracy for billions of
civilian and military users

Satellite ground tracks

30+satellites:~25 operational and up to 8 spares

GPS receiver

User component

Ground stations

control component

Ground stations

Track the GPS satellites
Monitor their transmissions
Perform analyses
Send commands and data to the
satellite constellation
“The current Operational Control
Segment (OCS) includes a master
control station, an alternate
master control station, 11
command and control antennas,
and 16 monitoring sites.

Trilateration using GPS satellites
is similar to triangulation

Triangulation measures angles between points of
known distance (i.e., a circle with a known radius).
Trilateration measures distance using the time it
takes a signal to travel from points of known
location.

Trilateration

Satellite Ranging
GPS is based on satellite ranging, i.e. distance from
satellites
Satellites are precise reference points; we determine
our distance from them
We will assume for now that we know exactly where
the satellites is and how far away from it we are.
If we are lost and we know that we are 11,000 miles
from Satellite 1, this means we are somewhere on
the surface of a sphere whose middle is Satellite 1
and whose radius is 11,000 miles
11k mi

Trilateration

If we also know that we are 12,000 miles from
Satellite 2 we can narrow down where we must
be only place in universe is on a circular plane
where these two spheres intersect
If we also know that we are 13,000 miles from
Satellite 3 our situation improves immensely.
There are only 2 points in universe where all
three three spheres intersect.
(one of them is not on earth)

A Simplified Process

u Working Assumptions:
Orbital positions of the satellites can be accurately computed with respect to the earth at
any time
GPS receiver on the ground can measure the distance between a receiver and a satellite for
at least three satellites at the same time
Receiver location can be defined using three coordinates: latitude, longitude, altitude
We can write three equations that relate the three distance observations to the known
coordinates of the satellites and the unknown coordinates of the receiver.
Three equations can be solved for the three unknowns (lat, long, alt).

So, what else do we nee to know to measure the
distance to a GPS satellite?

Timing signals transmitted by the satellites travel at the speed of light to receivers on
the ground.
The exact moment of signal transmission at the satellite and the exact moment of
signal reception at the receiver’s antenna must be accurately registered in order for
the distance and, consequently, the position calculation to be accurate.

So how do we measure the distance to a GPS
satellite?

Time Difference
The GPS receiver compares the time a signal was transmitted by a satellite with the
time it was received. The time difference tells the GPS receiver how far away the
satellite is.
Calculating Distance
Velocity x Time = Distance
Radio waves travel at the speed of light, roughly 186,000 miles per second

So how do we measure the distance to a GPS
satellite?

An Example
Radio waves travel at the speed of light, roughly 186,000 miles per second
If the GPS unit in my hand sends a signal that is received by an orbiting satellite 0.06 seconds
later, how far away is the satellite?
186,000 mps x 0.06 seconds = 11,160 miles
Clock Accuracy
0.061 seconds = 11,346 miles
-0.060 seconds = 11,160 miles
A difference of 0.001 seconds = 186 miles of error!
Clock accuracy is critical to the operation of a GPS!

The Problem

An Example

Satellites: Very expensive, highly accurate atomic clock
GPS receiver: inexpensive, therefore less accurate clock
Timing error occurs as the arriving satellite signal is timed at the
receiver
We have a fourth unknown in our system of equations

So, we actually need FOUR satellites to
accurately determine:

Latitude
Longitude
Altitude
Timing Error

Other sources of error:

Delays passing through atmosphere (refraction)
Solid objects (e.g., trees, mountains), obstructing the signal path
Solid objects (e.g. buildings) reflecting signals

Other sources of error

Satellite may not be where reported (orbital errors – very
uncommon)
- Satellites too close together (orbital geometry – very
common

A note on the value of basic science

• Some rocks glow
• Space is bendy
• Time is wibbly-wobbly

All GPS tech rests on some very basic
discoveries

“Cesium clocks are the most accurate ... and serve as
the primary standard for the definition of
the second in the metric system. By definition,
radiation produced by the transition between the two
hyperfine ground states of cesium has a frequency of
exactly 9,192,631,770 Hz.” (Wikipedia)
1 second = 9,192,631,770 vibrations of a cesium atom
First cesium atomic clock: 1955

All GPS tech rests on some very basic
discoveries

Theory of General Relativity, published 1915
Postulated the occurrence of time dilation due to
the influence of gravity on space-time
The closer you are to a large body like the Earth
the slower time runs, thus time runs slower for
someone on the surface of the earth compared
to satellites in orbit around the earth
u TL;DR: Gravity slows down time

But wait.

Uncle Al’s Theory of General Relativity:
“...relies on Riemannian geometry, which was proposed in the 19th
century to explain how spaces and curves interact – but dismissed as
derivative and effectively useless in its time.

The Problem

How do we represent a spherical object (planet earth) on a flat surface (a map)?
Need to accurately preserve the shape and spatial arrangement of geographic
features
In practice, this is not possible for every feature at the same time.
So, a wide variety of map projections have been developed to accurately
represent:
Area
Shape
Distance

Direction
Each projection does something well (while doing several other things poorly)

How do you choose a projection?

If your map requires that a particular spatial property to be held
true, then a good projection must preserve that property.
A good projection minimizes distortion in your area of interest.

Geographic Coordinate Systems:
How to locate a point on the earth

A geographic coordinate system uses a three-dimensional spherical surface to
define locations on the earth.
Components:
Angular unit of measure (parallels)
Meridian definition
Datum/spheroid

Geographic Coordinate Systems:
How to locate a point on the earth

geographic coordinate system uses a three-dimensional spherical surface to
define locations on the earth.
Parallels: latitude
(measures north-south)
Distances are measured in degrees (Degrees, Minutes, Seconds or Decimal Degrees)
Meridians: longitude (measures east-west)
Datum/spheroid: describes the shape of the earth

Components of Geographic Coordinate Systems:
Parallels (Angular unit of measure

In a geographic coordinate system, a point is referenced by its longitude and
latitude values.
Longitude and latitude are angles measured from the earth's center to a point
on the earth's surface. The angles often are measured in degrees.

Components of Geographic Coordinate Systems:
Meridian definition

The line of zero longitude is called the prime meridian.
For most geographic coordinate systems, the prime meridian is the
longitude that passes through Greenwich, England.
Other countries use longitude lines that pass through Bern, Bogota, and
Paris as prime meridians

Components of Geographic Coordinate Systems:
Datum/Spheroid

Estimating Shape of the Earth: (It’s not perfectly round!)
The shape and size of a geographic coordinate system's surface is defined by a
sphere or spheroid.
The shape of the Earth is best modeled using a spheroid...
The N-S diameter of the planet is roughly 1/300 less than the E-W diameter

Components of Geographic Coordinate Systems:
Datum/Spheroid

Datum - Defines the surface (e.g. radius for a sphere, major axis and minor
axis) and the position of the surface relative to the center of the earth.

Components of Geographic Coordinate Systems:
Datum/Spheroid

While a spheroid approximates the shape of the earth, a datum defines the
position of the spheroid relative to the center of the earth.
A datum provides a frame of reference for measuring locations on the surface
of the earth

Components of Geographic Coordinate Systems:
Datum/Spheroid

Datums are defined using a network of
ground control points
NAD 83 is based on the adjustment of
250,000 points including 600 satellite
Doppler stations which constrain the system

Components of Geographic Coordinate Systems:
Datum/Spheroid

There may be infinite reference surfaces.
Nations or governing bodies can agree on points and surfaces as standard
references.
The most commonly used datums in North America are:
North American Datum (NAD) 1927 using the Clarke 1866 spheroid
NAD 1983 using the Geodetic Reference System (GRS) 1980 spheroid
World Geodetic System (WGS) 1984 using the WGS 1984 spheroid

Summary:
Geographic Coordinate Systems

A geographic coordinate system uses a three-dimensional
spherical surface to define locations on the earth.
Components:
Angular unit of measure
Meridian definition
Datum/spheroid

Projected Coordinate System

Projected coordinate systems are the solution to the problem of
representing a spherical object (the earth) on a flat surface (a
map)
A projected coordinate system is defined on a flat, two-
dimensional surface.
Projected coordinate systems are always based on a geographic
coordinate system that is based on a sphere or spheroid.
Components:
Linear unit of measure
Map projections (math to convert from 3D -> 2D)
Parameters for conversion

Projected Coordinate Systems:
Converting from 3D to 2D

Representing the earth's surface in two dimensions causes
distortion in the data
Diff. projections = diff. distortions
Some projections are designed to minimize the distortion of
one or two of the data's characteristics.
e.g. A projection could maintain the area of a feature but alter
its shape

A warning

You will often find yourself working with many data layers from many different sources
Arc will display layers using “on the fly” reprojection, but this is for display only; the
underlying data may still have mismatching projections
u It is your job to verify the projection information in each layer. It is good to get in the
habit of doing this first.
Conducting analyses on layers with differing projections
will produce errors. Do
not
trust “on the fly” reprojections.
Tip:
Decide on a projection at the start of the project.
Your employer/client might have strong feelings
about which you use.
Often you can use the projection of your
most important data layer and re-project
subsequent layers to match

State Plane

The State Plane Coordinate System (SPS or SPCS) is a set of 124 geographic zones or
coordinate systems designed for specific regions of the United States. Each state contains
one or more state plane zones, the boundaries of which usually follow county lines.
It uses a Cartesian coordinate system to specify locations instead of latitude and longitude.
By using the Cartesian coordinate system's simple XY coordinates, "plane surveying"
methods can be used, speeding up and simplifying calculations.
State Plane is highly accurate within each zone (error less than 1:10,000). Outside a
specific state plane zone accuracy rapidly declines, thus the system is not useful for
regional or national mapping

Note:

While its fun to beat up on the Mercator projection, remember that
all map
projections include some amount of distortion of at least one of the following
(usually more than one):
Area
Shape
Distance
Direction
So, pick a projection that minimizes these distortions within
your area of interest.

Working with map projections

These different projections are all “correct.”
But if your project relies on a stack of data layers that are all using different
projections and you analyze them as though they are all the same, you’re
gonna have a bad time.
Arc is happy to let you do this. Your results will be wrong and you may not discover
it until much later (if at all).

GIS Data Models

Vector data model: A representation of the world using geometry (points,
lines, and polygons which are described using equations). Vector models
are useful for storing data that has discrete boundaries, such as country
borders, land parcels, and streets.
u Vector data consists of individual points, which (for 2D data) are stored as
pairs of (x, y) coordinates. The points may be joined in a particular order to
create lines, or joined into closed rings to create polygons, but all vector
data fundamentally consists of lists of co-ordinates that define vertices,
together with rules to determine whether and how those vertices are
joined.
u Note: whereas raster data consists of an array of regularly spaced cells, the
points in a vector dataset need not be regularly spaced.

GIS Data Models

Raster data model: A representation of the world as a surface divided into a regular grid of cells. Raster models are useful for storing data that varies continuously, as in an aerial photograph, a satellite image, a surface of chemical concentrations, or an elevation
surface.
Raster data is made up of pixels (or cells), and each pixel has one value associated with it.

Raster/Vector Comparison

In many/most cases, both vector and raster representations of the same data are
possible:
ProTip: No matter how far you zoom in on a vector layer, its edges will always appear
smooth, but eventually a raster’s cells/pixels will become visible

Vector vs Raster: How to choose

Do you want to work with
pixels or coordinates?
Raster data works with
pixels. Vector data consists
of coordinates

Vector vs Raster: How to choose?

What is your map scale?
Vectors can scale objects
up to the size of a
billboard. But you don’t
always get that type of
flexibility with raster data

Vector vs Raster: How to choose?

Do you have restrictions
for file size? Raster file size
can be larger than vector
data sets showing the
same phenomenon and
area. Vector file sizes
depend on the number
and complexity of features.

Rasters

We can also use a vector data
model to show elevation
data
This map includes both
vector and raster data

Benefits of Raster Data

Useful for storing continuous or rapidly changing variables
Overlay and buffering tools are computationally simple and rapid
to perform
Allows us to use “Map Algebra” in which maps are used as part
of algebraic expressions
Careful use of raster algebra can often produce results much,
much faster than the equivalent vector workflow.

Drawbacks of Raster Data

Tradeoffs between precision and storage space
Precision determined by cell size
Making cell size smaller increases precision, but also
exponentially increases required storage space.
Each raster cell can store only one value

Rasterization

Converts a vector layer (polygon, lines, points) to a raster layer
Conceptual Algorithm
Lay the raster grid on top of the vector layer
for each raster grid cell, check to see whether it overlaps
“enough” w/ a feature in the vector layer
if so, assign one of the feature’s attributes to the raster cell

A note on raster resolution:

In map algebra, normally, raster layers should have the
same extent, and the same cell size.
See example of issue to the right
It is recommended that you reproject the
raster directly before performing the
analysis.

Rasterization: Cell Assignment Type

CELL_CENTER:
• If a cell overlaps more than 1 feature, choose the one w/ the largest area
within the cell
MAXIMUM_AREA:
• If there is more than one feature in a cell with the same value, combine all
the features w/ the same value and choose the one w/ the largest
combined area

Simple Raster Analysis

Map Algebra
Resample
Reclassify
Mask
Slope

Map Algebra

(Like the Field Calculator for every pixel)
Raster layers may be combined through operations – addition,
subtraction and multiplication
Example: Change in Rainfall

Resample

If working with multiple rasters with different
resolutions (cell size), it is best to RESAMPLE
them to the same resolution.
Resampling will change cell size by the extent of
the original raster will be retained
Two methods for calculating values of new cells:
Interpolation- uses values of the surrounding
cells to calculate the value of the new cell
Aggregation- combines the individual values
from a groups of cells to produce a single,
coarser cell with a new value

Resample

Changing raster cell size can only decrease the amount of data present in the layer
Decreasing cell size increases resolution (more, smaller pixels), but the values in these new cells are based
on the data in the original layer (interpolated from original data)
Increasing cell size decreases resolution (fewer, larger pixels) by aggregating (sum, average, max, majority,
etc..) the values in multiple cells into a singe new cell

Reclassify

Changes the values in a
raster:
Grouping values (ex. right)
One-to-one mapping of
values

Grouping Values

Group original data via these classification methods:
manual
equal interval
defined interval
quantile
natural breaks (Jenks)
standard deviation

Reclassify: Using Booleans

Sometimes you want to find areas that meet multiple criteria
We can do this by:
Reclassifying a set of rasters to only contain T/F values based
on your criteria
Only grid cells (i.e. pixels) where all values are true

Mask Rasters

Masks are raster layers where all cells have a value of 1 or 0
In this case, the cell values (1,0) represent logical values
(TRUE/FALSE) rather than data values
Mask layers can be used to select/extract values from an input
raster
This is what we were doing with the silly example a few slides
ago, but the tool is extremely powerful for analyzing raster data
Extract by Mask tool

Calculating Slope

A cell’s slope is affected by its neighbors
For each cell:
The Slope tool calculates ‘rise over run’ from C’s
value to each its 8 neighbors and picks the
maximum increase
This tool identifies the steepest uphill ascent (in
degrees)for a given position

Calculating Slope

Slope cannot be calculated for cells that don’t have 8 neighboring
cells (e.g. edge cells)
NOTE: The output raster has been trimmed by 1 cell all around

Calculating Slope

Slope can potentially be useful in identifying landscape patterns
and processes.
Examples?
Watershed delineation
Erosion risk
Estimating stormwater
retention/run off
Mapping roads/trails