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1 Assessment
On the geodetic model and its solution

1 Assessment
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The following are some recommended self-assessment questions for the lesson called On the geodetic model and it’s solution. We covered much of this in class, but it also requires an understanding of the key concepts in the two supplementary readings.

a) Generally, why did we say we need to know gravity when dealing with geomatics networks?

b) Imagine you have to convince a surveyor who works in the field with a theodolite that this is true. Give him or her three specific examples of why the gravity field is important in geomatics networks. In each case, be sure to include the level of error that might be committed if gravity is ignored.

c) Imagine that you have to convince someone who uses the GPS receiver in their phone that this is true. Provide examples again.

Among the outputs of a GPS system is the height of the antenna.

a) What heights are obtained from GPS?

b) Above (or below) what reference surface are these heights?

c) Why can’t we just use those heights in areas of application such as geomatics networks? Give at least one specific example.

d) So then why do we use the reference surface in b)?

e) What is the geoid? Use the word “equipotential” (or the words “equal potential”) in your answer.

f) What equation allows us to relate the heights coming from a GPS system to those referred to the geoid?

a) What is ?

b) About how accurately can we compute it today? (Keep in mind that the reading I gave you on the related fundamental concepts was written in 1993.)

c) Approximately what is its value at your current location? I would recommend using an online calculator like this (as described here) to answer this question. You may also find interesting and relevant Natural Resources Canada’s page on the modernization of their height reference.

In class we discussed a few cheap ways (e.g. costing less than a dollar) of determining the true direction of the gravity vector.

a) Can you name and describe one of them?

b) That vector is exactly perpendicular to what reference surface?

c) What are the names and symbols of the angles that define the relationship between that vector and the equivalent vector in the reference surface used by GPS. Provide the component in the north-south direction and the component in the east-west direction.

d) About how big are these values typically? And in extreme cases?

e) What two coordinate systems are related to each other by these same angles?

f) Using what you know from your coordinate frames class, can you write the full relationship between a position vector in each of these coordinate frames?

Draw a sketch showing the following:

- the surface of the earth
- the plumb line
- the globally best-fitting geoid
- the gravity vector
- the globally best-fitting ellipsoid
- the normal gravity vector
- the height of the geoid
- the total deflection of the vertical

What is the role of the so-called geoid undulations and deflections of the vertical in precise:

a) 1D network adjustments?

b) 2D network adjustments?

c) 3D network adjustments?

The second paper you read does some pretty great work propagating the errors that can be caused by ignoring gravity when establishing a geodetic network. You don’t have to be able to do this yourself of course, but I want you to be able to interpret its results. For example, what maximum error you would expect to see if the distance between two points A and B is reduced to the ellipsoid where A and B are some 50 km apart and the geoid undulation in the area is roughly 30 m at both points?

We’re going to look in more detail in a later topic at reducing measurements made at the surface of the earth down to the ellipsoid. But for now I want you to tell me what the second paper you read says is the level of accuracy needed in gravimetrically predicted geoid undulations, , and deflections of the vertical, and , for the purpose of reducing geodetic observations.

a) What is the full (general) form of the unknown parameter vector in integrated geodesy? What do each of the terms mean?

b) Why do we need to apply a constraint when solving the usual linearized functional models in geomatics networks?

c) What is that constraint in the general case? Provide an intuitive explanation of what each term means.

d) What is the constraint in the case of the least squares solution?

e) Why do we call it a “least squares” solution?

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