4. Minimizing Errors

Now that we know where errors come from and how they behave, how do we deal with them in our measurements?

a. Mistakes

The only way to know if a mistake exists is to repeat the measurement. Having just two measurements which differ greatly only indicates that there is a mistake, it doesn’t tell us which measurement has the mistake. To find out, we need at least a third measurement. Hopefully the third one will be close to one of the first two. If it is, we’ve found our culprit and would toss the wrong measurement. If it’s not, then we have a dilemma on our hands - evidently we’re doing something wrong (think Target (c) or (d)).

How much of a difference is acceptable before a mistake is presumed? It depends on a number of things such as what’s being measured, what type of equipment is used, etc. Pacing a distance would have a higher mistake threshold than measuring the distance with a total station. In the old days, measuring distances with a 100 ft steel tape required the crew to keep a tally of full tape lengths. A common mistake was "dropping a tape" which showed up when the forward and back distances differed by about 100 feet. An obvious tally mistake. But what if the distances differed by about one foot?

We need to identify and eliminate mistakes because, as we’ll see in a bit, if we include those measurements, the mistake is spread into all the measurements degrading our good ones.

b. Systematic Errors

As we saw when introducing this error type, we can rid ourselves of them either mathematically or, in many cases, procedurally.
When you input the temperature and pressure in your total station, you are providing the information it needs to mathematically compensate for atmospheric effect on distance measurement. You may also have it set to compensate for earth curvature and refraction for longer distances. How about that prism offset?

In some cases, we don’t need to know the amount of systematic error if we can just get rid of it. Equipment maladjustment is a generally a systematic error. Being surveyors, and a clever group at that, we use specific measurement procedures which allow those maladjustments to cancel.

Example

A collimation error in a level means that the line of sight is not truly horizontal when the bubble is centered, Figure C-2. That means each rod reading will be too high or too low.

We can collimate the level to determine the error amount.

Once we have that, we can either adjust the level to eliminate the collimation error or we can mathematically apply it to correct a reading.

Or, because the error is a function of distance, we can balance our backsight (BS) and foresight (FS) distances and the error will go away:

 

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Figure C-2
Compensating Systematic Error Procedurally

  

 As you learn equipment use, you will be taught specific procedures which help compensate systematic errors.

c. Random Errors
Using appropriate equipment under favorable conditions by an experienced field crew and repeating measurements is the best way to minimize random errors. This is why we spend so much time familiarizing ourselves with our equipment and measuring so many times.


An example of angle measurement standards are the precise traverse theodolite and angle specifications in the FGCS Standards and Specifications for Geodetic Control Networks. These are summarized in Tables C-1 and C-2.

Table C-1

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Table C-2
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Both tables demonstrate that to achieve higher accuracy, finer resolution equipment is needed along with additional measurements. The general relationship of random error magnitude and number of measurements is shown in Figure C-3.

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Repeating a measurement several times can result in a larger initial random error reduction. Adding more measurements further reduces error.

Eventually a point of diminishing returns is reached: additional measurements don't appreciably lower the error. It's up to the surveyor to determine when that point is reached based on the job and equipment available.

This relationship is non-linear so doubling measurements doesn't cut error in half. Plus the graph never reaches 0 error because there is always some error present.

Figure C-3
Repeated Measurements and Error
 

 

Because random errors are “small” and tend to cancel with repeated measurements we analyze them statistically. This is where we encounter terms like “standard deviation”, “95% confidence interval”, “least squares”, “rejection limit from the mean”, etc. We’ll look at this basic analysis in the next chapter.