Assessment Progress:

The following are some recommended self-assessment questions for the lesson called Let’s linearize our general functional model.

We’re going to focus in much of our course on geomatics networks on parametric models and adjustments. As such, the goal of these questions is to provide a quick review of the general case and some important elements of the other types of models.

Question 1:

Determine whether each of the following is linear or nonlinear, and explain why or what rule(s) you use to make such an assessment. The unknowns are in the \mathbf{x} vector and the measurements are in the \mathbf{l}_{true} vector.

a) 5\mathbf{x} = \mathbf{l}_{true}

b) \pi\mathbf{x} + \mathbf{l}_{true} = \mathbf{0}

c) \mathbf{l}_{true}^2 = \mathbf{x}

c) \mathbf{x}\mathbf{l}_{true} = \mathbf{0}

Question 2:

a) What, again, are our three general functional models? Name them and show how we write them as a model.

b) What are the linearized forms of the three general functional models? Define each of the terms used and indicate the sizes of any matrices and vectors.

c) What are the degrees of freedom in each case?

Question 3:

a) What are \mathbf{e} and \boldsymbol{\delta}? And will we ever know them?

b) In the literature (and possibly in your other courses), what is another variable often used in place of \mathbf{e}? Does it matter which we use?

c) What role do \mathbf{e} and \boldsymbol{\delta} play in arriving at the linearized forms of our general models? Without doing the derivation again, show me what you mean here, e.g. by recalling Taylor’s Theorem.

Question 4:

a) Derive and write out the functional model for an observed height difference between two stations.

b) Use the work of this lesson to write out the linearized form of the same model. Be as rigorous with your notation and the steps used as I have been with mine. (No need to worry about datum or height issues here. Just assume a constant datum for the problem.)

Question 5:

Can you determine and linearize the model for the following simple leveling network? Since you’re going to get lots of practice with parametric models and their linearization in this course, I want you to do this one in the form of a condition model. Assume point 1 is a known benchmark and that we seek the heights of the other three points. And assume all measurements are taken once, where \Delta h_{AB} = h_Bh_A.