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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.