Multiple T-test

In the Norwegian market, it's common to use a method based on a multiple T-test.

What is a Gross Error?

In adjustment theory, it's common to assume that errors larger than three times the standard deviation are gross errors. It's also common to base adjustment theory on the assumption that measurement errors are random and normally distributed. According to this, one can expect that approximately 0.27% of measurements will be classified as gross errors.

In practice, gross errors in measurement data occur more frequently than they should according to the normal distribution. There are typically between 1% and 5% gross errors in a typical measurement dataset.

Without a gross error detection routine, one could to some extent locate gross errors by examining residuals. However, this is a cumbersome and unreliable method, as gross errors often distribute across many residuals and are therefore difficult to locate.

Undetected gross errors affect the unknowns calculated in the adjustment. The ability to eliminate such gross errors is motivation enough to perform gross error detection. Equally important is that gross error detection gives us an estimate of how large the remaining gross errors are. Remaining gross errors are errors that are not determined with sufficient certainty for the observation to be discarded. By knowing the size of these errors, we are able to quantify the maximum effect on the unknowns and on derived quantities in the control network. This forms the basis for evaluating external reliability.

Free or Constrained Adjustment?

We can distinguish between two types of gross error detection:

1. Gross error detection after free adjustment
2. Gross error detection after constrained adjustment

The same routine is applied, but the results must be interpreted differently.

Warning

After constrained adjustment, we do not know whether the located gross errors are actual gross errors or an effect of constraints in the control network. We must therefore not discard observations after a gross error detection based on constrained adjustment!

Exceptions to this rule exist in cases where we know with great certainty that the control points contain only negligible errors. An example of this is when the control points consist of modern first-order points. Gross error detection after constrained adjustment is well-suited for locating major errors in data, such as whole meters, whole gons, typographical errors, and so on.

After free adjustment, we are allowed to discard observations that we can assert with great certainty contain a gross error. But of course, we must also here first search thoroughly for the cause of the error.

Theoretical Description of the Method

We attempt to estimate a gross error in each observation, but only one at a time. This is done by introducing a gross error parameter in the error equation matrix, which must then be expanded with a column. The column gets a value of one in the row for the observation in question. If the observation is a direction, the Schreiber equation of the set must also get a value of one.

The figure shows how the error equation matrix is expanded with a column for the gross error parameter.