The distinction between measurement and observation

We know that all practical measurements are subject to errors. Let’s think again of our measurement model, for example:

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

We like to think that after pre-processing and pre-adjusting the measurements we’ll be left with the desired signal, l_{measured}, plus only random or stochastic errors, n, i.e. as we have discussed, pre-processing and pre-adjustment will remove everything except the stochastic errors, n, as reflected in the figure below where I have shown the related parts of the process in yellow and labeled the distinction between measurement and observation.

Let’s assume (for now) that we can do this, and refer to the term l_{true}+n as our observation. And think about the observation as the value or quantity going into the estimation process, i.e. while we haven’t until now, we’re now going to start distinguishing between measurements and observations:

measurement = observation + systematic errors + gross errors

which allows us to write:

observation = measurementsystematic errorsgross errors

This is also reflected on the right side of the arrows between the yellow boxes in the figure below.


The notion of a stochastic model

Crudely put, the estimation process (shown by the box in the above figure labeled “adjusting/estimating” and implemented by our parametric least squares adjustment equations) can be thought of as a system that takes in observations and spits out the desired parameters. But it won’t work without an unambiguous and sufficiently representative understanding of the stochastic errors that are still present in the observations.

So for very practical reasons we need to make some assumptions about the statistical properties of those observations. Let’s think about them as stochastic random variables, meaning that they differ from the true or underlying quantity by some random random statistical variation as we discussed above.

And, let’s assume that an estimate of that statistical variation is fully defined by some parameters that are available to us a-priori, i.e. in advance of our survey.

Together, these parameters and the underlying assumptions are called the stochastic model, i.e. the stochastic model describes the errors in our observations.

It includes variables that are considered to be fixed, i.e. constant during the adjustment (either known or predetermined) and variables that are considered free, i.e. those to be determined by the adjustment.1

The next lesson covers the key statistical concepts you need to know in order to understand and apply stochastic models to applications that require geomatics networks.