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Alpha GPS & Geodesy offers online courses on the mathematical and geodetic foundation of GPS/GNSS (Global Positioning System / Global Navigation Satellite System). The depth of coverage and the mathematical presentation provide the detailed knowledge necessary to understand techniques and algorithms for positioning at all levels of accuracy. These courses are available for credit and non-credit through Michigan Technological University (MTU).



Course Delivery

Students access the courses via the CANVAS Learning Management System. The lecture videos are fully supported with graphics and audio. There are many Mathcad solutions with audio that use real data to amplify key topics. All solutions are original implementations, including complete GPS Precise Point Positioning and GPS baseline solutions using actual pseudorange and carrier phase observations. Selected Mathcad implementations, i.e. worksheets with source code, are available to students for further experimentation and in-depth development beyond class requirements. There is no commercial GPS/GNSS or geodetic software used.

Students’ progress and understanding of the material is assessed by multiple choice tests. The test questions are in form of images which makes it possible to incorporate mathematical expressions and graphics to address the details of the material to be tested. Courses that are taken for credit require additional laboratory work. 

Skype is the prime means of communication between the instructor and the students. Using simple screen sharing allows for effective online communication and individualized learning. Students can contact the instructor at any time during regular working hours. The class size is ONE student. Therefore, the students learning experience is that enjoyed when taking a private classes.


Credit Options

Courses labeled GPS XXX are a unique series offered by Alpha GPS & Geodesy. They are available for non-credit, and anybody can enroll at any time during the year. If enrollment takes place during a regular MTU semester, the course terminates at the end of the following semester. There is an implicit understanding that students are familiar with college-level calculus, statistics, and matrix operations. The students are free to decide how much effort they wish to put into these courses. If a minimum total score is achieved, a letter of completion can be requested. For registration please check GPS-GAP Studies at Michigan Technological University.

Courses labeled SU XXXX  are available for credit through Michigan Technological University as part of the Integrated Spatial Technology MS Program. Students not currently enrolled at Michigan Technology University (MTU) can enroll as non-degree student to get college credit. Tuition, semester schedule, prerequisites, grading and other matters strictly follow MTU rules. For more information visit the MTU website. To register as a non-degree student at MTU please click here. Prior contact with the instructor is recommended.


Textbook  (all courses)

GPS Satellite Surveying

Alfred Leick, Lev Rapoport, Dmitry Tatarnikov
Edition, 2015, 800 pages
Publisher: J. Wiley
& Sons
ISBN 978-1-118-67557-1

Errata (4th Edition)

Errata (3rd Edition)  Request from Author
Errata (2nd Edition)  Request from Author


GPS 101 Mathematics Refresher (non-credit, MTU, no course fee if taken concurrently 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.

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

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.

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.

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.

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.

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.

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  

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.

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.

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.