The adjustment of the observations Surveying II. The adjustment of the observations of a single quantity

A short revision Total error: the difference between the true value and the observation: The total error (ei) can be subdivided into two parts, the systematic error (di), and the random error (xi): system. error random err.

A short revision The probability density function The PDF of a normally distributed probabilistic variable:

A short revision The „3-sigma” rule: where: a – is the expected value of the x probabilistic variable

In order to compute ei the true value should be known. A short revision The mean error of the variable can be computed using the error (Gauss): . In order to compute ei the true value should be known. Thus the mean error must be estimated from the observations: Number of redundant observations (degree of freedom)

The relationship between the weight and the mean error: A short revision The relationship between the weight and the mean error: ,

Accuracy vs. Precision The result is accurate but not precise. The result is precise but not accurate. Precision: The value is close to the true value (total error is low) Accuracy: The approach gives similar results under changes conditions, too. (the mean error computed from the observations is low)

Adjusting the observations of a single quantity „A single observation is not an observation…” In case of more observations, discrepancies are experienced: Let’s assume, that the observations are statistically independent, the observations are free of systematic error : the mean error of the individual observations are known:

Adjusting the observations of a single quantity Task: to remove the discrepancies from the observations and to compute the most likely value of the quantity. Corrections are applied to each observation: The adjusted value of the observations (no system. error) Question: How could the correction values be determined (infinite number of possible correction sets) Let’s minimize the corrections!

Adjusting the observations of a single quantity Usually the weighted square of the corrections are minimized: „least squares adjustment” Let’s combine the two equations:

Adjusting the observations of a single quantity Let’s find the minimum of the function: thus:

Adjusting the observations of a single quantity By reordering this equation: Note that this formula is the formula of the weighted mean: an undistorted estimation; the most efficient one;

Adjusting the observations of a single quantity Let’s check the computations: The corrections are computed for each observation: Let’s compute the following sum: Check!

Adjusting the observations of a single quantity Task: The computation of the mean error of the adjusted value. The law of error propagation can be used: Thus the mean error of the adjusted value is:

Adjusting the observations of a single quantity The mean error of the adjusted value: Introducing the relationship between the weight and the mean error:

Adjusting the observations of a single quantity Thus the mean error of the adjusted value can be computed by: Please note: It is necessary to know the mean error of the unit weight before the computation of the mean error (‘a priori’ mean error of unit weight) In this case the computed mean error values are based on variables, which are known before the adjustment process. These are the ‘a priori’ mean error values. They can be used for planning the observations.

Adjusting the observations of a single quantity After the adjustment, the mean error of the unit weight can be estimated using the following equation: where f is the number of redundant observations (degree of freedom) In case of n observations of a single quantity f=n-1: This equation uses quantities available after the adjustment process. ‘a posteriori’ mean error of unit weight

Adjusting the observations of a single quantity Using the ‘a posteriori’ (after the adjustment) mean error of unit weights, the mean error of the observations as well as the adjusted value can be computed:

Adjusting the observations of a single quantity Question: How much is the weight of the adjusted value? Since the mean error of the unit weight is known, using the relationship between the weight and the mean error, the weight of the adjusted value can be computed: Thus the weight of the adjusted value equals to the sum of the weights of the observations.

The ‘a priori’ and the ‘a posteriori’ mean error the ‘a priori’ mean error reflect our knowledge before the observations (instrument specifications, prior experiences); the ‘a posteriori’ mean error can be computed after the adjustment of the observations (experienced mean error); in both cases the mean error of the unit weight can be computed; when the two values are close to each other, then: the ‘a priori’ mean error values are realistic (our observations are accurate enough) our observations are not affected by blunders

The process of adjustment Given observations and mean err.: L1, L2, L3, …, Ln and m1, m2, m3, …, mn Let’s choose an ‘a priori’ mean error of the unit weight: m Define the weights of the observations: Compute the adjusted value:

The process of adjustment Compute the corrections: Check the adjustment: Compute the ‘a posteriori’ mean error of unit weight: Compute the weight of the adjusted value: Compute the ‘a posteriori’ mean error of the observations and the adjusted value:

When the ‘a priori’ mean error of observations are equal In this case: Thus the observations have a unit weight! The adjusted value: The corrections:

When the ‘a priori’ mean error of observations are equal The ‘a posteriori’ mean error of unit weight: The weight of the adjusted value: The ‘a posteriori’ mean error of the observations and the adjusted values:

A simple example L1 = 121.115m ± 10mm L2 = 121.119m ± 5mm The results of the distance observations between two points are given with their mean error values: L1 = 121.115m ± 10mm L2 = 121.119m ± 5mm L3 = 121.121m ± 5mm L4 = 121.118m ± 10mm L5 = 121.116m ± 10mm

A simple example L0= 120,110 Li [mm] mi mi2 Pi piLi vi Pivi Vi2 Pivi2 +5 10 100 1 +4 16 +9 5 25 4 +36 +11 +44 -2 -8 +8 +1 +6 +3 9 S 11 +99 42 Let’s choose an ‘a priori’ mean error of u.w.: m=5 The adjusted value: The ‘a posteriori’ mean error of u.w.: