## Courses

GPS 101 Mathematics Refresher
(non-credit,
MTU, no course fee if taken concurrent with another GPS
XXX course): *Real number systems and operations, basic functions,
linear algebra with emphasis on matrices, eigenvalues, eigenvectors,
differentiation, integration, elements of vector calculus, spherical
trigonometry, interpolation, solution of linear and nonlinear
equation systems, quadratic forms, minimization of multi-dimensional
functions, basic univariate distributions and functions thereof,
multivariate normal distribution.*

This refresher course serves to practice mathematical skills. The
course *selectively*
addresses topics that are needed for understanding the mathematics
of positioning with GPS/GNSS, including adjustments. The material covered
is typically spread over several traditional university courses.

GPS 401 Adjustments with Observation Equations
(non-credit,
MTU): *Errors, stochastic and mathematical models,
quadratic forms, linearization and variance-covariance propagation
of multi-dimensional nonlinear functions, least-squares algorithm of
observation equations, position estimation using surveying and GPS
vector measurements that are nonlinear functions of parameters;
review of statistics and linear algebra. Equivalent of 1 Cr *

This is the first of three adjustments courses and focusses on the
*observation*__ __
*equation*__ __
*model*. Since most
functional relationships of interest are nonlinear, we deal with the
*linearization* of
multidimensional and multivariate functions in detail. The
*stochastic*__ __
*model* expressing the quality of the observations is quantified
by the variance-covariance matrix. Since one often undervalues the
importance of the stochastic model, it is stressed in this unit. The
*mathematical*__ __
*model, *
which expresses the
functional relationship between the observations and the
parameters, is developed for various applications. We also focus on the application-independent aspects
of the
adjustment algorithm. This unit concludes with
*variance-covariance*__
__*propagation*. We
assume that the distributions of the random variables exist and are
specified by their mean and variance covariance matrix. We do not
deal with specific types of distributions in this unit. The
*chi-square test*, which is
fundamental in testing the validity of the least-squares solution,
is presented without derivation.

GPS 402 Adjustment Algorithms
(non-credit,
MTU): *Error ellipses and ellipsoids, propagation of
estimated quantities, a priori information on parameters, adjustment
of implicitly related observations and parameters, mixed model,
condition equation model, sequential solutions, testing conditions
on nonlinear parametric functions. Equivalent of 1Cr. Recommended
prerequisite GPS 401.*

The *mixed model*, which
deals with implicit functions of observations and parameters, is at
the center of this unit. Specifying the mixed model allows us to
derive the *condition*__
__*equation model*. The
latter model contains
no parameters.
Starting with the mixed model we develop the
*sequential solutions* for
all models. Additional specifications yield the models that
allow incorporation of *a
priori*__ __*information
on parameters*, or the testing of conditions between parameters.
We consider nonlinear conditions between parameters, and then apply
the *General Linear Hypothesis*
Model after the linearization.
*
Ellipses of standard deviation*,
often referred to as error ellipses for brevity, are introduced as a
simple form of higher-dimensional probability regions. The
*magnification* of these
regions to assure that a certain probability is included and the
shapes of these regions as a function of correlation will be
studied.

GPS 403 Quality Control with Adjustments
(non-credit,
MTU): *Geometry of least-squares, definition of
network coordinate systems, singularities, probability regions,
minimal and inner constraints, invariant quantities, multivariate
normal distribution, relevant statistical tests, type I/II errors,
internal and external reliability, absorption of errors, blunder
detection, decorrelation, inversion of patterned and large matrices,
numerical aspects; Kalman filtering. Equivalent of 1 Cr*.
Recommended prerequisite GPS 401.

One often thinks that "adjustments and high accuracy go together"
somehow. We will stress in this unit that adjustment is best viewed
as a tool for *objective
quality control, *and that the respective techniques and
rules apply to high-accurate or low-accurate applications. This unit
addresses statistics in detail as it applies to least squares. We
introduce the *multivariate
normal distribution* and derive the relevant statistical tests.
After discussing type-I and type-II errors, we introduce
*internal and external
reliability* as major tools of
*quality control*, followed
by a discussion of strategies for
*blunder detection*. This
includes recognizing the fact that least-squares solutions tend to
*absorb* parts of the
blunder, i.e. making residuals to reveal only the "visible part" of
the actual errors. Introducing
*minimal* and
*inner constraints* is a
general way of dealing with the
*invariance of observations*
with respect to the definition of the coordinate system. The effect
of different minimal constraints on the size and shape of the
probability regions will be discussed in the context of the
*over-all geometry* of an
adjustment as is implied by the stochastic and mathematical models.

GPS 441 Three-Dimensional Geodetic Model
(non-credit,
MTU): *Conventional celestial and terrestrial
references frames, precession, nutation, polar motion, geodetic
datum, geoid, ellipsoid of revolution, geodetic coordinates, height
systems, 3D geodetic model and model observations, reduction of
observations, observation equations, partial derivatives, 3D network
adjustments, height-controlled 3D networks, GPS vector observations,
review of spherical trigonometry and spherical harmonic expansions.
Prerequisite: GPS 401, GPS 403, Equivalent of 1 Cr* *
Recommended prerequisite GPS 401.*

The 3D geodetic model is the *
unified model* that applies to surveys and networks of __any__
size and shape. The popularity of this model stems from the
fact that it readily incorporates 3-dimensional GPS vector
observations. With accurate geoid undulations now widely available,
the transition from the traditionally "horizontal" and "vertical"
datums to the more natural 3-dimensional approach can finally be
made. This model deals with all observations, and is by far the
*simplest model*, at least mathematically speaking.
It is applicable to positioning with traditional
observations or with GPS. Common to all applications is the
*geodetic reference frame*.
This unit, therefore, includes details on the definition and
maintenance of the geodetic frame and on geophysical phenomena that
cause *temporal variation of
coordinates*. **
**

GPS 442 Ellipsoidal Surface Model
(non-credit,
MTU): *Geodesic line on the ellipsoidal surface,
geodesic curvature, differential equations of the geodesic, direct
and inverse solutions, 2D network adjustment on the ellipsoidal
surface, partial derivatives, reduction of observations, traditional
horizontal and vertical networks in surveying and geodesy; in-depth
review of differential geometry. Equivalent of 1 Cr*.
Recommended prerequisite GPS 441.

The ellipsoidal surface model deals with 2-dimensional computations
on the ellipsoidal surface. This model is viewed here as an
*intermediary but necessary*
model that eventually leads to the
*conformal mapping models*
discussed in GPS 443. The ellipsoidal model and the "horizontal"
datum are conceptually the same in that both refer to the
2-dimensional ellipsoidal surface. Computations on the ellipsoidal
surface require the *geodesic
line*, which considerably *
complicates* the mathematics. We carefully identify the
ellipsoidal surface model observables, and show how these are
obtained from those of the 3D geodetic model used in GPS 441. We
then formulate the network adjustment of an ellipsoidal surface
network.

GPS 443 Conformal Mapping Model
(non-credit,
MTU): *Conformal mapping of the ellipsoidal surface,
meridian convergence, point scale factor; State Plane Coordinate
systems, Transverse Mercator, Equatorial Mercator, Lambert Conformal
with one or two standard parallels, polar azimuthal, and UTM;
reduction of observations, computations on the conformal map and
relation to the surface of the earth; review of complex variables.
Equivalent of 1 Cr*.
Recommended prerequisite GPS 401, 441.

The popularity of the *State
Plane Coordinate Systems* (SPCS) among surveyors in the USA makes
the conformal mapping model very relevant. The SPCS are primarily
patchwork of transverse Mercator and Lambert conformal mappings
covering the country.
We look in detail at the meaning of
*conformality* and the
mathematical conditions that lead to such a property. We introduce
the *point scale factor*,
*line scale factor*, and
the *meridian*__ __
*convergence* and derive
the mapping equations that map the ellipsoidal surface to the
*conformal map* and vice
versa. We further derive and discuss the so-called
*mapping*
*elements* that allow us to
convert the 2D ellipsoidal surface model observations to conformal
mapping model observations, and adjust networks on the conformal
map.
The conformal mapping model is the most "abstract model", in the
sense that physical observations must first be reduced to the 3D
geodetic model, then to the 2D ellipsoidal model, and finally, to
the conformal mapping model. It is of course also important to apply
these reductions in the reverse order whenever angles and distance
on the map are to be used on the physical surface of the earth.

GPS 490 GNSS Receiver Antennas
(non-credit,
MTU, instructor is D. Tatarnikov): *Basics of
electromagnetic waves, polarization, antenna angular response
pattern and gain, polarization properties of GNSS user antennas,
phase pattern, phase center variations and antenna calibrations,
carrier phase multipath, reflections from the underlying terrain,
antenna down/up ratio, basics of transmission lines, antenna
mismatch and frequency response, cable losses, noise propagation and
signal-to-noise ratio. Equivalent of 1Cr*

GNSS receiver antennas are treated from the user's point of view.
The basics of electromagnetic field theory are discussed at the
level necessary for understanding antenna performance parameters.
Complex notations and dB scale are introduced as basic tools for
treating performance. The antenna angular response pattern is
presented for transmitting and receiving modes with reciprocity
theorem serving to connect these two modes of operation.
We introduce the antenna effective square. The topic of polarization
properties are addressed by using polarization transformations of
type linear, circular and elliptical. Polarization losses for the
receiver antenna are estimated.

The antenna phase center for GNSS applications is rigorously defined
as the reference point for positioning for the limiting case when
the satellite paths cover the whole top the semi sphere in a
continuous and homogeneous manner. The elevation mask is taken into
account. This approach serves as a base for the treatment of antenna
calibration procedures. Phase center variations are discussed as
phase pattern residuals when the pattern is being transformed to the
thus defined phase center.

The plane reflective surface model is used for estimating multipath
from underplaying terrain. The antenna down-up ratio is introduced
and the multipath error is being discussed for typical cases.

A detailed discussion of wave propagation and reflections in
transmission lines leads to the treatment of antenna mismatch and
frequency response. Typical sources of noise for GNSS applications
are analyzed, the propagation of signal and noise through the
electronic circuits is discussed in detail; the role of the low
noise amplifier (LNA) is stressed.

This course concludes with a section on estimating the signal to
noise ratio at the receiver input.

GPS 570 Fundamentals of Satellite Positioning
(non-credit,
MTU): *ITRF and ICRF references frames and
transformations, tectonic plate motions, precession, nutation, polar
motion, rotational and atomic time scales, GPS time, normal orbits,
Kepler's laws and equation, topocentric satellite motions,
visibility, perturbation of satellite orbits, solar radiation
pressure, impact of asymmetry of gravity field and earth's
flattening; GPS, GLONASS and Galileo satellite systems. Equivalent
of 1 Cr*.
Recommended prerequisite GPS 401, 441.

This unit introduces the *
geocentric motions of satellites* for the case of a spherically
symmetric gravity field. The three
*Kepler laws* and the
*Kepler equation* are
derived and applied to various orbits. We develop the expressions to
compute *topocentric azimuth, elevation angle, and
distance* of satellites and study
*satellite trajectories* as
a function of semimajor axis and inclination, and construct
visibility charts.
Our understanding of orbital motions and the limitations of
satellite ephemerides will be further deepened by considering the
effects of the *flattening of
the earth, asymmetries* in the gravity field, and *solar *
*radiation pressure* on the
orbits of satellites.
This unit introduces the precise definitions of conventional
*celestial* and
*terrestrial reference frames*,
including all physical phenomena that change coordinates of points
in an earth-centered and earth-fixed reference frame. These
foundations, which are also addressed in parts in GPS 441, are
important considering that we are now able determine geocentric
locations with centimeter accuracy.
An initial discussion on the GPS, GLONASS and planned Galileo
satellite systems will be given.

GPS 571 Precise Point Positioning
(non-credit),
MTU: *Pseudorange and carrier phase observables,
satellite time, relativity, broadcast and precise ephemerides, range
iteration, receiver and satellite clock errors; singularities,
tropospheric refraction and absorption, impact of the ionosphere,
solid earth tides, ocean loading, satellite antenna offset, phase
windup correction, closed form solutions; Kalman filter; timing,
mapping of the spatial and temporal variation of the troposphere and
ionosphere. Equivalent of 1 Cr*.
Recommended prerequisite GPS 401, 441, 570.

The *pseudorange* and
*carrier phase* equations
will be discussed term by term, including
*receiver and satellite clock
errors,* *phase ambiguities*,
*hardware delays* and
signal *multipath* at the
receiver and at the satellite. Considerable details will be provided
on the physics of the *
troposphere* and *
ionosphere* and their impact on pseudoranges and carrier phases.
The *point positioning
solution* using *the
broadcast ephemeris* and pseudorange observations to four and
more satellites from a single station will be given in both the
linearized and closed forms. We accurately compute the topocentric
distance, i.e. the distance the signal travels from the epoch of
transmission at the satellite to the epoch of reception at the
receiver antenna, taking the *
finite velocity of light* and the
*earth's rotation* into
account.
The core of this unit is Precise Point Positioning (PPP) using
dual-frequency pseudorange and carrier phase observations, the
precise ephemeris, and precise satellite clock corrections. Because
PPP potentially gives centimeter accurate geocentric coordinates it
is necessary to deal with *
solid earth tides* and *
ocean loading*. The *phase
windup* correction, resulting from the right circular
polarization of the satellite signals, does not cancel and must be
considered.

GPS 572 Precise Relative Positioning
(non-credit,
MTU): *Differencing observables in space and time,
common-mode error reduction, geometry-free solutions, widelaning,
closed-form solutions, cycle slips, constraint solutions, integer
ambiguity estimation, LAMBDA, antenna calibration, multipath on
pseudoranges and carrier phases, spatial vector networks,
differential corrections, global data collection and maintenance,
GPS services. Equivalent of 1 Cr*.
Recommended prerequisite GPS 401, 441, 571.

In relative positioning, either in the
*static or kinematic mode*,
we are using the popular *
single -, double -, or triple difference functions* to determine
the accurate relative location of two or more receivers and take
advantage of error cancellation/reduction when differencing the
observables.
The *geometry-free solutions*,
which do not parameterize the position of the receiver explicitly,
are suitable to discover and possibly remove cycle slips in
preprocessing.
A major part of this unit is devoted to the powerful
*LAMBDA* (least-squares
ambiguity decorrelation adjustment) technique to fix the estimated
ambiguities to integer. Whenever centimeter-accurate relative
positioning is needed, it is necessary to determine and fix the
integer values of the ambiguities.

**
SU 5020 Data Analysis and Adjustment**
(credit,
MTU).
This graduate course and is offered as part of the Program on
Geospatial Technology. It is designed for students who desire a
deep
understanding of adjustment. The course content is that of GPS 401, 402, and
403 listed above. 3 Cr; MTU
prerequisites and rules apply.

**SU 5021 Geodetic Models**
(credit,
MTU).
This graduate course is offered as part of the Program on
Geospatial Technology. It is designed for students who desire a
deep
understanding of geodesy. The courses combines the material of GPS 441, 442, and
443 listed above. 3 Cr; MTU
prerequisites and rules apply.

**
SU 5022 Positioning with GNSS **
(credit,
MTU).
This graduate course is offered as part of the Program on
Geospatial Technology. It is designed for students who desire a
deep
understanding of positioning with GNSS. The courses combines the
mateerial form GPS
570, 571, and 572 given above. 3 Cr;
MTU prerequisites and rules apply.

**
SU 5023 Geospatial Positioning**
(credit,
MTU).
This graduate course is offered as part of the Program on
Geospatial Technology. It is designed for
*interdisciplinary*
students who need to acquire a thorough understanding of
adjustments, geodesy, and positioning with GNSS in just one course. Therefore
the courses GPS 401, 403, 441, 443, 570, 571, and 572 are the basis
for this comprehensive course.
When deciding which material to include, preference was given to the
GNSS courses 570, 571, and 572. Note that the course GPS 442, which
deals primarily with the mixed adjustment model, has been excluded.
In addition, all material that requires advanced
understanding of Differential Geometry has been excluded.3 Cr; MTU
prerequisites and rules apply, which positively includes calculus, statistics,
and matrices.