Let’s improve our “big picture” view

You now know enough to revisit the big picture of what we’re doing in this course. So we’re going to do that from a few angles.

The measurement model revisited

When we looked at a simple form of the measurement equation before, we said that:

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

Using terms introduced since then (and sticking with the scalar parametric case to help make things easier to write out), the measurement can be written as l_{measured}, the observable as f(x), and the error as e. Which gives us:

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

With what you now know, e.g. from earlier mini courses and as we summarized in the opening remarks of this module, we can write the error term as follows:

    \begin{equation*} e = b + s f(x) + \alpha f(x) + \delta t + n + g \end{equation}

where, again:

b is a bias

sf(x) is a systematic scale factor error (due to the scale factor s)

\alpha f(x) is a systematic misalignment error (due to the misalignment \alpha)

\delta t is a systematic time-dependent error (i.e. a drift)

n is a random error (stochastic noise)

g is a gross error (to be avoided and/or detected)

From which, by straightforward substitution, we can write the following general form of the measurement model:

    \begin{equation*} \boxed{l_{measured} = f(x) - (b + s f(x) + \alpha f(x) + \delta t) + n + g} \end{equation*}

A “table of contents” for the course

I’ve taken the time to revisit the measurement equation here because it can provide a nice “table of contents” for this course – both what we have already seen and what we are soon going to see – if we rearrange it into four key terms as follows:

    \begin{equation*} (l_{measured} - f(x)) + (b +s f(x) + \alpha f(x) + \delta t) + n + g = 0 \end{equation*}

The first of these terms, (l_{measured} - f(x)), is a model that captures the relationship between the measurements and the observables. We’ve talked a lot about this relationship and f(x) in particular under the banners of functional modeling and linearization (including the work you did in the lab on functional modeling and linearization for the key observation equations). You should be thinking back to the general forms of the functional model. And to the approaches to coming up with the design matrix, \mathbf{A}, and misclosure vector, \mathbf{w}. In fact, you might even have noticed that the term (l_{measured} - f(x)) evaluated at the approximate values of the parameters is the misclosure vector.

Before leaving this first term, it might be worth emphasizing again that the function f(x) can be thought of as bringing the observables into measurement space in the parametric case. It can be thought of as a sort of “true” measurement – or what the value of the measurement would be if it was free or errors, e.g. f(x) = l_{true}. Which you might notice is the functional model itself for the parametric case.

The second term, (b +s f(x) + \alpha f(x) + \delta t), is a model for the so-called systematic errors. As you know, these are dealt with through a combination of:

  • calibration
  • differencing; and
  • (sometimes) estimation.

In the latter case – of trying to estimate these systematic error terms – you’d need to be able to develop and linearize a functional model that includes some of these systematic error parameters. I find the above form of the measurement model helpful in that regard because it expresses them in terms of the parameters, x.

The third term, n, is a way of representing the stochastic errors that we will seek to minimize through estimation procedures. Later in this module, we’re going to focus on using parametric least squares for this purpose. And then later, we’re going to consider the notion of stochastic modeling in detail as it will apply to estimation, to the pre-adjustment of observations, and to the post-adjustment of a network.

Finally, the last term, g, represents gross errors. We will treat these too, especially as it pertains to avoiding and detecting them as part of the pre-adjustment processes.

The process of geodetic modeling and estimation for geomatics networks

As we have discussed in class, an even better “table of contents” for this course might be provided by the following geodetic modeling and estimation process for geomatics networks. You can also download this in PDF format »

The following schedules are for Alex’s in-class students:

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