Courses - GPS 401
GPS 401 Adjustments with
Observation Equations (Catalog description): 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. Prerequisites: MAT 262, MAT 332, equivalent or
consent. Lec 1. Cr 1
This is the first of three
adjustments courses. Since in most cases the observations are an
explicit function of parameters, we focus on the observation
equation model. Because of the prevailing nonlinear functional
relationships, we deal with the linearization of multidimensional
and multivariate functions in detail. A simple example of a
nonlinear function is the equation expressing the distance in
terms of coordinates. The observation equation method is
particularly easy to implement because of the simple rule: for
each observation there is one observation equation.
At the core of each adjustment
are the stochastic and mathematical models. The stochastic model
contains the information about the quality of the observations and
is quantified by the variance-covariance matrix. Users frequently
overlook or undervalue the importance of the stochastic model. The
mathematical model expresses the mathematical relationship between
the observations and the parameters. Often this relationship is
difficult to describe mathematically due to the various physical
influences on the observations. Examples are the effects of
refraction on angles or the troposphere and ionosphere on GPS
signal propagation. In the adjustment units we deal with model
observations that have a simple mathematical relation to the
parameters in order to have the time to focus on
application-independent aspects of adjustments. The concepts of
mathematical model and model observations are dealt with in the
units GPS 441 – 443 in detail.
This adjustment unit also deals
with variance-covariance propagation of functions of random
variables. We assume that the distributions of the random
variables exist and are specified by their mean and variance
covariance matrix. We do not emphasize 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.
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