Precision Survey & Deformations Monitoring

Precision survey and def. mon.

Unimore Università degli Studi di Modena e Reggio Emilia

Filippo Ribes

NoteWave_RF

Settore: ICAR/06

Theory of observation error

Autore degli appunti: Filippo Ribes

Gli appunti sono stati scritti sulla base delle lezioni svolte dalla Professoressa Cristina Castagnetti e delle sue dispense. Per dubbi, chiarimenti o altro, mi trovi su Instagram: ig: NoteWave_RF ig: fil_ribes

Accuracy limitation of modern equipment

1. Atmospheric and target conditions:
   • Atmospheric refraction caused by temperature, pressure, humidity ⇒ distances are shortened or increased.
   • Very important: The intervenibility between targets and instruments.
   • The temperature effect is highest in modern instruments.
2. Equipment, design and precision:
   • Calibration and testing are essential for correct working.
   • The electronic equipment requires more calibration than others.
3. Instrument operation factor:
   • Now surveyors are more focused on data processing and analysis.
   • One/two persons are enough because the instruments work automatically ⇒ efficiency is increased and costs are saving.

Error properties of measurements

Measurement = measurement best estimate + measurement uncertainty

3 Types of errors:

1. Blunders or gross errors (mistake or failure of the instruments or surveyors).
2. Random errors (accidental errors: they can be estimated or minimized but not eliminated).
3. Systematic errors (known as "bias", a type of error that remains the same, it's possible to remove with math models).

Definition

Observable: physical or geometrical quantity to which a numerical value can be assigned through a measurement process with a degree of certainty.

Measurement or Observation: numerical value that is assigned to an observable. It is also used to refer to the actual process of assigning a numerical value to an observable.

The exact values of observables cannot be determined, but estimated. Also, the "Fundamental principle of measurement in surveying" says that no measurement is exact and the true value of an observable can never be known. The exact values of the observables exist, but they cannot be determined exactly.

Precision: the degree of closeness of one measurement to another one. It increases when random errors decrease.

Accuracy: the degree of closeness of one measurement to its true value.

One-dimensional variable

To determine a variable x means to provide an estimate of the variable based on m observations.

_^x̄ = (x₁ + x₂ + ... + xₘ)/m → _^x̄ = (∑ xᵢ)/m (i=1 to m) → mean

σ² = (∑ (xᵢ - _^x̄)²)/(m-1) → variance

σₓ̄² = ∑ᶦⱼ₌₁ (xⱼ - _^x̄ⱼ)² / m(m-1) → σₓ²/m = variance of the mean

Standard deviation serves as a measure of precision of measurement. We can determine the precision of measurement by using a weight (p_i).

m-dimensional variable

Expressed by means of several one-dimensional variables. Covariance is a numerical measure of the correlation between variables that defines a m-dimensional variable. Variance-covariance matrix expresses the errors associated with the variables and their relationships.

Properties:

• Symmetric matrix
• Invertible matrix

Correlation:

• ρ = 0 : uncorrelated variables
• ρ = +1 : variables with positive correlation
• ρ = -1 : variables with negative correlation

Error propagation laws

Consider a variable z as a function of two variables x and y as expressed by equation z = f(x, y). The variance of variable z (being σ_z²) is to be determined given the variances of x as σ_x², variance of y as σ_y² and the covariance of x and y as σ_xy(= σ_yx):

σ_z² = [ ^∂f(x,y)⁄_∂x ]² σ_x² + [ ^∂f(x,y)⁄_∂y ]² σ_y² + 2 [ ^∂f(x,y)⁄_∂x ] [ ^∂f(x,y)⁄_∂y ] σ_xy

In matrix form: