A quick step back for perspective on what we’ve been doing

In our early days together, we looked at the simple problem of estimating the width of a hallway from a bunch of measurements taken with a tape measure, and we began discussing the following general form of the error model:

    \begin{equation*} \text{measurement }=\text{ observable }+\text{ error} \end{equation*}

In that case we were directly measuring the desired parameter (the width, i.e. a distance), meaning that \text{ observable}=x and we could write:

    \begin{equation*} \l_{measured}=x+e \end{equation*}

Following that we quickly recognized that we don’t always have the luxury of directly measuring the parameter we’re after, so we grew up in our thinking to the following:

    \begin{equation*} l_{measured}=f(x)+e \end{equation*}

which is meant to express the observable as some more general function of the parameter, \text{ observable}=f(x), so that that component of our measurement model could be brought into “measurement space”.

With this in hand we went further to write the error as follows:

    \begin{equation*} e = l_{measured} - f(x) \end{equation*}

and the residual as follows:

    \begin{equation*} r = l_{measured} - f(\hat{x}) \end{equation*}

And although this still pretty much holds for the parametric case that we’re going to concentrate on for most of the rest of this course, we were forced to grow up in our thinking again in order to recognize the kinds of relationships that exist between observables and measurements in geomatics engineering.

This is precisely what the whole topic of functional modeling was about, for example, which yielded the following general cases:

    \begin{equation*} \mathbf{F}(\mathbf{x},\mathbf{l}_{true})=\mathbf{0}\text{ for the combined model} \end{equation*}

    \begin{equation*} \mathbf{l}_{true} - \mathbf{F}(\mathbf{x})=\mathbf{0}\text{ for the parametric model} \end{equation*}

    \begin{equation*} \mathbf{F}(\mathbf{l}_{true})=\mathbf{0}\text{ for the condition model} \end{equation*}

where the left sides of these equations could be substituted in for f(x) above for the general multivariate cases.

By now you should be very good at deriving the general functional model for any relevant situation and then linearizing it. For the general case this means: 1) knowing which of the above forms applies; 2) being able to express and then linearize it; 3) deriving the misclosure vector \mathbf{w} and the design matrices \mathbf{A} and (if needed) \mathbf{B}; and 4) writing it all out in the form \mathbf{A}\boldsymbol{\delta} + \mathbf{e} + \mathbf{w} =\mathbf{0}.

And then, after recognizing that the parametric model is the star of the geomatics networks show, we focused our attention on that form and got really good both at developing those functional models and linearizing them.

At this point in the course I expect that you’d be able to do this for all of the following cases, whether using a parametric model or not:

  • Distances
  • Distances and other errors such as a scale factor
  • Azimuths
  • Anges
  • Simple trilaterations
  • Simple intersections
  • Coordinate transformations
  • Height differences, e.g. as in the case of leveling

This module

In this mini course we’re going to take things a step further. First, we’re going to revisit a few things that help us define and better understand the “big picture” as a bridge between what was summarized above and what’s coming in the following.

Second, we’re going to consider the notion of a geodetic model and what it means to consider the role of gravity in geomatics networks.

Then we’re going to look at the general approaches to solving the broader geodetic modeling problem before focusing on a recap of the process of least squares estimation which will form the basis of our solution finding in this course.

Directions

Just work your way through the lessons below.