Technical User Manuals for Survey Instrument Calibration

$$ D_{ij}=x_j-x_i-zpc\ \ \ for\ j > i\ \ \ (Eq. 2.1)$$

$$ D_{ij}=x_i-x_j-zpc\ \ \ for\ j < i\ \ \ (Eq. 2.2)$$

$$ \begin{matrix}D_{12}=&{+X}_2&&&-zpc\\D_{13}=&&{+X}_3&&-zpc\\D_{14}=&&&{+X}_4&-zpc\\D_{21}=&{+X}_2&&&-zpc\\D_{23}=&-X_2&+X_3&&-zpc\\D_{24}=&-X_2&&{+X}_4&-zpc\\D_{31}=&&+X_3&&-zpc\\D_{32}=&-X_2&{+X}_3&&-zpc\\D_{34}=&&-X_3&{+X}_4&-zpc\\D_{41}=&&&+X_4&-zpc\\D_{42}=&-X_2&&+X_4&-zpc\\D_{43}=&&-X_3&+X_4&-zpc\\\end{matrix}\ \ \ (Eq. 2.3)$$

$$ \left[\begin{matrix}D_{12}\\D_{13}\\D_{14}\\D_{21}\\D_{23}\\D_{24}\\D_{31}\\D_{32}\\D_{34}\\D_{41}\\D_{42}\\D_{43}\\\end{matrix}\right]=\left[\begin{matrix}1&0&0&-1\\0&1&0&-1\\0&0&1&-1\\1&0&0&-1\\-1&1&0&-1\\-1&0&1&-1\\0&1&0&-1\\-1&1&0&-1\\0&-1&1&-1\\0&0&1&-1\\-1&0&1&-1\\0&-1&1&-1\\\end{matrix}\right]\left[\begin{matrix}X_2\\X_3\\X_4\\zpc\\\end{matrix}\right]\ \ \ (Eq. 2.4)$$

$$\left[E_j,N_j\right]=\ Polar2Reclangular\left(Hz_{ij},d_{ij}\right)\ \ \ (Eq. 2.5)$$

$$ O_j=d_{j}sin\left(A_{1n}-A_{j}\right)\ \ \ (Eq. 2.6) $$

3.1.1 Zero Point Correction (zpc)

All distances measured by a particular EDM instrument and reflector combination (EDMI) are subject to a constant error caused by the following factors:

• Electrical delays, geometrical detours and eccentricities in the instruments
• Differences between the electronic centre and the mechanical centre of the instrument
• Differences between the optical and mechanical centres of the reflector (prism)

This errors may vary with changes of reflector, or after jolts, or with different instrument mountings. Zpc is an algebraic constant applied directly to every measurement.

3.1.2 Scale Correction Factor (scf)

Scale Correction Factor is proportional to the length of the distance measured and is caused by:

• Internal frequency errors, including those caused by external temperature and instrument “warm up” effects
• Errors of measured temperature, pressure and humidity affect the velocity (and path?) of the electromagnetic wave
• Non-homogeneous emissions/reception patterns from the emitting and receiving diodes (phase inhomogeneities)

3.1.3 Cyclic Errors

The precision of an EDM instrument dependents on the precision of the internal phase measurements. Unwanted interference through electronic/optical cross talk or multi-path effects of the transmitted signal on to the received signal causes cyclic errors. The major form of the cyclic error is sinusoidal with a wavelength equal to the unit length of the instrument. The unit length is the scale on which the EDM instrument measures the distance and is derived from the fine measuring frequency. Unit length is equal to one half of the modulation wavelength.

$$ IC=zpc+scf*D+1C\sin{\left(2\pi\frac{D}{U}\right)+2C\cos{\left(2\pi\frac{D}{U}\right)+3C\sin{\left(4\pi\frac{D}{U}\right)+4C\cos{\left(4\pi\frac{D}{U}\right)}}}}\ \ \ (Eq. 3.1) $$

$$ v_i=zpc+scf*D^C_i+1C\sin{\left(2\pi\frac{D_i}{U}\right)+2C\cos{\left(2\pi\frac{D_i}{U}\right)+3C\sin{\left(4\pi\frac{D_i}{U}\right)+4C\cos{\left(4\pi\frac{D_i}{U}\right)}}}}+(D^C_i-D_i)\ \ \ (Eq. 3.2)$$

$$D_i^C\ \ \ = Certified\ baseline\ distance $$

$$ V = AX + W\ \ \ (Eq. 3.3)$$

$$ V=\left[\begin{matrix}v_1\\v_2\\...\\v_n\\\end{matrix}\right]\ \ \ (Eq. 3.4) $$

$$ A=\left[\begin{matrix}1&D_1&\sin{2\pi\frac{D_1}{U}}&\cos{2\pi\frac{D_1}{U}}&\sin{4\pi\frac{D_1}{U}}&\cos{4\pi\frac{D_1}{U}}\\1&D_2&\sin{2\pi\frac{D_2}{U}}&\cos{2\pi\frac{D_2}{U}}&\sin{4\pi\frac{D_2}{U}}&\cos{4\pi\frac{D_2}{U}}\\...&...&...&...&...&...\\1&D_n&\sin{2\pi\frac{D_n}{U}}&\cos{2\pi\frac{D_n}{U}}&\sin{4\pi\frac{D_n}{U}}&\cos{4\pi\frac{D_n}{U}}\\\end{matrix}\right]\ \ \ (Eq. 3.5)$$

$$ W=\left[\begin{matrix}D_1^C-D_1\\D_2^C-D_2\\\ldots\\D_n^C-D_n\\\end{matrix}\right]\ \ \ (Eq. 3.6) $$

$$ X=\left[\begin{matrix}zpc\\scf\\1C\\2C\\3C\\4C\\\end{matrix}\right]\ \ \ (Eq. 3.7) $$

$$ p=\left[\begin{matrix}p_1&0&0&0\\0&p_2&0&0\\...&...&...&...\\0&0&0&p_n\\\end{matrix}\right]\ \ \ (Eq. 3.8)$$

$$ p_i=\frac{1}{S_{D_i}^2}\ \ \ (Eq. 3.9)$$

4.1.1 Meteorological instrumentation corrections

All meteorological instruments (thermometer, barometer, hygrometer) can reference calibration certificates to apply corrections to raw readings. These corrections are sourced from the most recent instrument calibration certificate current prior to the date of pillar survey. Calibration corrections are applied as zero point correction (zpc) to the raw meteorological instruments readings:

4.1.2 EDMI correction

EDMI calibration corrections are sourced from the most recent instrument calibration certificate current prior to the date of baseline survey:

If the meteorological corrections have not been applied to EDMI readings prior to upload, atmospheric calibration corrections will be applied prior to calculating the first velocity correction.

4.2.1 First velocity correction (K)

Where:

  K = First velocity correction in metres

  P = Pressure in millibars/hectopascals

  t = Dry temperature in degrees Celsius

  d = Distance in metres

  e = Partial water pressure in hectopascals

$$C = (N_{REF} – 1)10^6$$

  n_REF = Reference refractive index (Section 4.2.2)

$$D = (n_G – 1) 10^6 (273.15 / 1013.25)\ \ \ \ (Eq. 4.2)$$

  n_G = Group refractive index of atmosphere for standard conditions

Partial vapour pressure is derived from relative humidity:

Where:

  e = Partial water vapour pressure in hectopascals

  h = Relative humidity in percent

  E = Saturation water vapour pressure (hPa) at the dry bulb temperature:

Where:

  p = Atmospheric pressure in hectopascals

  t = Dry temperature in degrees Celsius

The terms C and D as well as EDM type, Unit Length, Modulation Frequency, Carrier Wavelength and Refractive Index are instrument specific (manufacturers supplied). Recommended values for common instruments are provided in the EDM Specifications Lookup Table. This table has been populated with information from the following sources:

• Rüeger, J. M. 1996. Electronic Distance Measurement - An Introduction, 4th corrected edition
• Rüeger. J. M. 2007a. First Velocity Correction in EDM: How to get the Constants C and D, Azimuth
• https://www.land.vic.gov.au/surveying/services/equipment-calibration-services
• CR Kennedy

These parameters are assigned to the EDM model and are used for all instruments of that model associated with the company. The Manufacturers Uncertainty Constant, Manufacturers Parts Per Million Uncertainty and Manufacturers Uncertainty Coverage Factor are required fields. These are used by Medjil to perform the statistical test recommended in ISO 17123-4:2012 (see Section 7).

Medjil treats phase and pulse measuring instruments differently (first velocity correction is different) thus the pulse instruments must have mets applied before importing data to Medjil.

Solving for cyclic errors can be performed for both types of instruments and is an option when performing and EDMI Calibration.

When solving for cyclic errors, Medjil will use a t-student test for both the first and second order cyclic errors. If errors are statistically insignificant, the design matrix will be compiled for either six, four or two parameters.

The Unit Lenth is a required field. It is used when test for cyclic errors, plotting the Return Phase Angle of Observed Distances plot at the end of the EDMI Calibration report and for calculation of the manufacturers reference refractive index when it has not been supplied in this form (See Section 4.2.2).

The Frequency, Carrier Wavelength, Manufacturers Reference Refractive Index, C Term and D Term are optional fields. These are used for calculating the atmospheric corrections for instruments. However, Medjil will refuse to process raw data where the atmospheric corrections have not been applied prior to upload, if insufficient parameters are supplied to calculate the atmospheric corrections. The C Term and D Term are used to calculate the first velocity correction (See Section 4.2.1). If the C or D Term are not supplied, Medjil uses the carrier wavelength, frequency and Manufacturers Refractive index to calculate these terms.

The Measurement Increments are a required field. Medjil will use this value for propagation of uncertainty when the pre-selected Uncertainty Budget requests instrument rounding is included from the Instrument Register Record – Uncertainty Budget Sources (See Section 5.2).

4.2.2 Reference Refractive Index

The reference refractive index (n_REF) is instrument specific and it is preferred that the manufacturer supplied value is used rather than calculated as the rounding error of the calculated value can be significant. If the refractive index is not provided Medjil uses Unit Length, Carrier Wavelength and Modulation Frequency stored in the instrument register to calculate this value.

Where:

  C_O = Velocity of light in a vacuum = 299 792 458 m/s (SI definition)

  λ_MOD = Carrier wavelength of the instrument

  f_MOD = Modulation frequency of the instrument

  U = The half of λ_MOD , called the unit length of the instrument.

4.2.3 Group refractive Index of light in the atmosphere

In electro-optical EDM, the refractive index is dependent on the wavelength of the visible or infrared radiation. Different frequencies have the same propagation velocity in a vacuum, but not in air because of the interference occurrence between the different frequencies. The signal resulting from the sum of all frequencies will have the so-called group velocity, which is always smaller than the phase velocities of its individual frequencies.

Resolution 3 from the International Association of Geodesy (IAG) in 1999 at its XXIIth General Assembly in Birmingham, made the following recommendations.

1. Subparagraphs (a) and (b) of Resolution No. 1 of the 13th General Assembly of IUGG (Berkeley 1963) be cancelled
2. The group refractive index in air for electronic distance measurement to better than one part per million (ppm) with visible and near-infrared waves in the atmosphere be computed using the computer procedure published by Ciddor & Hill in Applied Optics (1999, Vol. 38, No. 9, 1663-1667) and Ciddor in Applied Optics (1996, Vol. 35, No. 9, 1566-1573)
3. The following closed formulae be adopted for the computation of the group refractive index in air for electronic distance measurement (EDM) to within 1 ppm with visible and near-infrared waves in the atmosphere:

The following closed formulae relates to recommendation three for the computation of the group refractive index in air for electronic distance measurements:

Where:

  N_L = Group refractivity of visible and near infrared waves in ambient moist air. Valid for atmospheric conditions described by t, p and e

  n_L = Corresponding group refractive index

  N_G = Group refractivity index for standard conditions (visible light in dry air at 0 C, 1013.25 hPa, partial water vapour pressure = 0 and the air contains 0.0375 % CO₂). See Equation 4.7.

  t = Dry bulb temperature of air (C).

  p = Atmospheric pressure in hectopascals

  e = Partial water vapour pressure (hPa)

Where:

  λ = carrier wavelength in micrometres (μm) in a vacuum These closed formulae (Eq. 4.6 and 4.7) deviate less than 0.25 ppm from the accurate formulae between -30 C and +45 C, at 1000 hPa pressure, 100% relative humidity (without condensation) and for wavelengths between 650 nm and 850 nm. The 1.0 ppm stated in recommendation 3 of resolution 3 makes some allowance for anomalous refractivity and the uncertainty in the determination of the atmospheric parameters.

Recommendation 2 and 3 have been applied in the Medjil software where atmospheric corrections have not been applied to the uploaded EDM observation data. The formulae for recommendation 2 are used for calibrating a baseline. The formulae for recommendation 3 are used for calibrating EDM Instrumentation.

$$ d_R\ =\ \sqrt{\frac{\left(d_O\right)^2-\left(O_1-O_2\right)^2-(\Delta H)^2}{\left(1+\left(\frac{H_1-H_{REF}}{R+H_{REF}}\right)\right)\left(1+\left(\frac{H_2-H_{REF}}{R+H_{REF}}\right)\right)}}\ \ \ \ (Eq. 4.8) $$

$$ d_X\ =\ \sqrt{\left(d_R^2\left(1+\left(\frac{H_1-H_{REF}}{R+H_{REF}}\right)\right)\left(1+\left(\frac{H_2-H_{REF}}{R+H_{REF}}\right)\right)+(\Delta H)^2\right)+\left(O_1-O_2\right)^2}\ \ \ \ (Eq. 4.9) $$

$$ \mathrm{R\ }=\ \sqrt{pv}\ \ \ (Eq. 4.10) $$

$$ \mathrm{p\ }=\ \frac{a\left(1-e^2\right)}{(1-e^2{sin}^2{Q})^{3/2}}\ \ \ (Eq. 4.11) $$

$$ \mathrm{v\ }=\ \frac{a}{\sqrt{\left(1-e^2{sin}^2{Q}\right)}}\ \ \ (Eq. 4.12) $$

$$e^2\ =\ 2f-f^2\ \ \ (Eq. 4.13) $$

$$ C_{SU}=√(∑_{(i=1)}^n u_{i}^2)\ \ \ (Eq. 5.1) $$

$$ C_{EU}=C_{SU}k\ \ \ (Eq. 5.2) $$

$$ k=t_{(95\%, v_{eff})}\ \ \ (Eq. 5.3) $$

$$ v_{eff}=\frac{s^4}{C_{su}}\ \ \ (Eq. 5.4) $$

$$ s=∑_{(i=1)}^n\frac{u_{i}^4}{v_i}\ \ \ (Eq. 5.5) $$

5.2.1 Custom uncertainty budgets

The Medjil ‘Default’ uncertainty budget has all options in the 'Instrument Register Record - Uncertainty Budget Sources' and 'Derived - Uncertainty Budget Sources' selected. It also defines six 'Custom - Uncertainty Budget Sources' that are appropriate for typical calibrations.

When creating a Custom uncertainty budget, Medjil will use the Default as a template to initiate the creation of the new uncertainty budget.

Where:

Name = Custom uncertainty budget name. Uncertainty budgets are identified by a name that must be unique.

Company = The owner of the new uncertainty budget. The new budget(s) are accessible to any user within that company.

Std Dev Used When Statistically Zero = This value is used to overruled (overwrite) any standard deviation equal to zero. For example Medjil uses the sample standard deviation of observation sets composed of measurements between the same pillars to calculate an uncertainty of the EDMI measurements. When all measurements in the one observation set are the same or if there is only one measurement in the observation set, the sample standard deviation is mathematically zero.

Type = Definition of uncertainty type. Type A means a statistically derived component or uncertainty where Type B stands for any other derivation of uncertainty e.g. calibration report, manufacturer's specification or an estimate based on experience.

Distribution = Definition of distribution associated with the uncertainty. A normal distribution is most associated with continuous univariate distributions such as many random variables. A rectangular distribution represents an arbitrary outcome that lies between certain bound (equal probability anywhere within the range). For example rounding of a physical quantity or a resolution of digital instrument.

Uncertainty = The expanded uncertainty at 95% confidence in the units specified for the uncertainty source.

K = The coverage factor is used to expand the uncertainty from 68% confidence level to 95% confidence level. Typically k=2.0 for a normal distribution or sqrt(3) for a rectangular distribution. Medjil will auto complete this field as the distribution drop-down is changed but the user still has the ability to overwrite this field.

Dof = Degrees of freedom. For a Type A uncertainty this is equal to the number of measurements minus the number of unknowns (redundant measurements). For a Type B uncertainty Dof needs to be estimated. The following can be used as a guide: 3 for not very confident, 10 for moderate confidence, 30 for very confident.

Group = Uncertainty sources are categorised into 11 different groups. The group selected for an uncertainty source will affect the following:

• The combining of error sources in the calibration report
• The unit selection for this uncertainty source
• The propagation (contribution) of the error sources to the total uncertainty of the calibration

EDM scale factor = Uncertainty related to the EDM scale factor can be specified in parts per million (ppm), percent (%), scalar(x:1) or a ratio (1:x). All error sources belonging to this group will be combined before the contribution to the total uncertainty of the calibration according to the following formula:

Where:

  su_d = Standard uncertainty of the distance

  D = Distance

  csu_g = Combined standard uncertainty of the group

The 'Default' Uncertainty Budget lists an error source with the description 'EDM scale factor affected by Temp (Type B)'. The EDM scale factor can be affected by temperature. This uncertainty component is a Type B estimate based on prior knowledge and experience.

EDMI measurement = Uncertainty relating to the EDMI measurement can be specified in metres (m), millimetres (mm), parts per million (ppm), percent (%), scalar(x:1) or a ratio (1:x). All error sources belonging to this group will be combined before the contribution to the total uncertainty of the calibration according to the following formula:

Where:

  su_d = Standard uncertainty of the distance

  csu_g = Combined standard uncertainty of the group

EDM LS zero offset = Uncertainty relating to the EDM Least Squares zero offset can be specified in metres (m) or millimetres (mm). All error sources belonging to this group will be combined before the contribution to the total uncertainty of the calibration according to the following formula 4.13

Temperature = Uncertainty related to the temperature can be specified in degrees Celsius (°C) or degrees Fahrenheit (°F). All error sources belonging to this group will be combined before the contribution to the total uncertainty of the calibration according to the following formula:

Where:

  su_d = Standard uncertainty of the distance

  D = Distance

  csu_g = Combined standard uncertainty of the group

  K = Partial differential for temperature defined as:

Where:

  p = Pressure [hPa]

  t = Temperature [C]

  n_G = Manufacturers reference refractive index

  e = Partial water vapour Pressure [hPa] defined as:

Where:

  E = Saturation water vapour [hPa] at the dry bulb temperature

  h = Relative humidity [%]

Pressure = Uncertainty related to the pressure can be specified in millibars (mBar), hectopascals (hPa) or millimetres of mercury (mmHg). All error sources belonging to this group will be combined before the contribution to the total uncertainty of the calibration according to the following formula:

Where:

  su_d = Standard uncertainty of the distance

  D = Distance

  csu_g = Combined standard uncertainty of the group

  L = Partial differential for Pressure defined as:

Humidity = Uncertainty relating to the humidity must be specified as a percent (%). All error sources belonging to this group will be combined before the contribution to the total uncertainty of the calibration according to the following formula:

Where:

  su_d = Standard uncertainty of the distance

  D = Distance

  csu_g = Combined standard uncertainty of the group

  M = Partial differential for humidity defined as:

Certified distances = Uncertainty relating to the certified distances can be specified in metres (m), millimetres (mm), parts per million (ppm), percent (%), scalar(x:1) or a ratio (1:x). All error sources belonging to this group will be combined before the contribution to the total uncertainty of the calibration according to the following formula:

Where:

  su_d = Standard uncertainty of the distance

  csu_g = Combined standard uncertainty of the group

EDMI calibration = Uncertainty relating to the EDMI calibration can be specified in metres (m) or millimetres (mm). All error sources belonging to this group will be combined before the contribution to the total uncertainty of the calibration according to the Eg. 5.21.

Centring = Uncertainty relating to the centring can be specified in metres (m) or millimetres (mm). All error sources belonging to this group will be combined before the contribution to the total uncertainty of the calibration according to the Eg. 5.21.

Heights = Uncertainty relating to the heights can be specified in metres (m) or millimetres (mm). All error sources belonging to this group will be combined before the contribution to the total uncertainty of the calibration according to the following formula:

Where:

  su_d = Standard uncertainty of the distance

  D = Distance

  csu_g = Combined standard uncertainty of the group

  ∆H = Height difference between the from and to pillars

Offsets = Uncertainty relating to the offsets can be specified in metres (m) or millimetres (mm). All error sources belonging to this group will be combined before the contribution to the total uncertainty of the calibration according to the following formula:

Where:

  su_d = Standard uncertainty of the distance

  D = Distance

  csu_g = Combined standard uncertainty of the group

  ∆O = Difference in offset between from and to pillars

5.2.2 Uncertainty sources in reports

If the 'Default' uncertianty budget is used, Medjil automatically populates uncertainty sources shown in Table 5.1 and Table 5.2 when processing the calibration of EDM Baselines and EDM Instrumentation respectively. Except for the following, these uncertainty sources are optional when using a custom uncertainty budget:

• EDMI Calibration - Uncertainty of LSA EDMI correction
• EDM LS zero offset - ZPC uncertainty
• Certified Distances - LSA uncertainty

Table 5.1: Medjil uncertainty budget for calibration of EDM Baseline


Table 5.2: Medjil uncertainty budget for calibration of EDM Instrumentation

The automatically populated uncertainty sources are defined as follows:

• Certified distances - Least Squares Adjustment (LSA) uncertainty - The uncertainty of each inter-pillar distance is uniquely assigned from the diagonal elements of the CoFactorMatrix derived from the least squares adjustment of the baseline calibration. The degrees of freedom is equal to the degrees of freedom of the calibration of the baseline and the coverage factor is derived from t-student distribution at 95% and selected degrees of freedom t_(95%,dof)
• EDM LS zero offset - Zero Point Correction (ZPC) uncertainty - The uncertainty of the zero point correction is assigned from last element of the vCoFactorMatrix derived from the least squares adjustment of the baseline calibration. The degrees of freedom is equal to the degrees of freedom of the calibration of the baseline and the coverage factor is the t-student at 95% and selected degrees of freedom (t_(95%,dof))
• EDM scale factor - EDMI Reg13 Scale correction factor - The uncertainty, degrees of freedom and coverage factor is assigned from the values specified in the register of calibration certificates of EDM instrument (EDMI). The selected values correspond to the most current certificate (at the time of survey) of the EDMI used for the calibration of the EDM Baseline
• EDM scale factor - EDM scale factor (drift over time) - The uncertainty is derived using linear regression of the scale factors from all available calibration certificates of the same EDM Baseline (certified distances). The scale factor is extrapolated at the time of calibration survey and the difference between this value and the last computed value (as per the most current certificate) is adopted as the uncertainty. This uncertainty source is assigned 30 degrees of freedom and a coverage factor of √3
• EDMI measurement - Distance Instrument Rounding - The uncertainty is equivalent to half of the instrument measurement increment reading. This uncertainty source is assigned 100 degrees of freedom and a coverage factor of √3.
• EDMI measurement - Linear regression on EDM distance standard deviations - Table 5.3 shows an example of a typical subset of EDMI measurements observed on a EDM Baseline. Measurements with the same ‘From’ and ‘To’ pillar are grouped into sets, then average and standard deviations of these sets are tabulated. The ‘Std Dev Used when statistically zero’ field value defined in the specified Uncertainty Budget of the calibration can be used to exclude and overwrite experimental standard deviations of measurement sets that have all the same values (eg. experimental standard deviations equal to zero).
  Linear regression is used on the average ‘slope distance’ and ‘standard deviation’ to derive a formula for the experimental standard deviation of EDMI measurements in the calibration: s_B = P mm + Q ppm. Where s_B is the standard deviation of a baseline interval.
  Uncertainties are uniquely assigned to each measurement set as calculated from the derived linear regression equation. This uncertainty source is assigned 30 degrees of freedom and the coverage factor is the t-student at 95% and 30 degrees of freedom t_(95%,30)
• Heights - Height differences - During the Calibrate an EDM Baseline process, a *.csv file is required to upload the standard uncertainty (standard deviation) of the heights of each pillar. These uncertainties are used to derive the uncertainty of the height differences according to the equation 5.6.
  $$ {uc}_{\Delta H}=k\sqrt{\left(H_i^2+H_j^2\right)}\ \ \ (Eq. 5.6) $$

  Where:

    uc_ΔH = Expanded uncertainty of the height difference between pillar i and pillar j

    uc_o = Expanded uncertainty of the offset between pillar i and pillar j

    H_i = Tabulated experimental standard deviation of the ‘From Pillar’

    H_j = Tabulated experimental standard deviation of the ‘To Pillar’

    k = t_(95%,30)

• Humidity - Hygrometer calibrated correction factor - Hygrometers used for baseline calibration surveys must have a calibration certificate that indicates the uncertainty of the readings. Medjil will automatically assign a current hygrometer calibration based on the survey date. The uncertainty, coverage factor and degrees of freedom specified for that hygrometer calibration record will be used for the uncertainty ‘Humidity - Hygrometer calibrated correction factor’
• Humidity - Hygrometer rounding - The uncertainty is equivalent to half the measurement increment reading of the instrument used for the calibration. This uncertainty source is assigned 100 degrees of freedom and a coverage factor of √3.
• Offsets - Offsets - Table 5.4 shows the results of a typical alignment survey. The offset of each pillar is surveyed from multiple pillars. The offset results with the same ‘To Pillar’ are grouped into sets, then the average and experimental standard deviations of these sets are tabulated. The uncertainty of each offset is derived from the standard deviation of the offset measurements at both the ‘From Pillar’ and the ‘To Pillar’.
  $$ {\rm uc}_o=k\sqrt{\left(o_i^2+o_j^2\right)}\ \ \ (Eq. 5.7) $$

  Where:

    uc_o = Expanded uncertainty of the offset between pillar i and pillar j

    o_i = Tabulated experimental standard deviation of the ‘From Pillar’

    o_j = Tabulated experimental standard deviation of the ‘To Pillar’

    k = t_(95%,30)

  This uncertainty source is assigned 30 degrees of freedom and the coverage factor is the t-student at 95% and 30 degrees of freedom t_(95%,30)

• Pressure - Barometer calibrated correction factor - Barometers used for calibration surveys must have a calibration certificate that indicates the uncertainty of the readings. Medjil will automatically assign a current barometer calibration based on the survey date. The uncertainty, coverage factor and degrees of freedom specified for that hygrometer calibration will be used for the uncertainty ‘Pressure - Barometer calibrated correction factor’
• Pressure - Barometer rounding - The uncertainty is equivalent to half the measurement increment reading of the instrument used for the calibration. This uncertainty source is assigned 100 degrees of freedom and a coverage factor of √3.
• Temperature - Thermometer calibrated correction factor - Thermometers used for calibration surveys must have a calibration that indicates the uncertainty of the readings. Medjil will automatically assign a current thermometer calibration based on the survey date. The uncertainty, coverage factor and degrees of freedom specified for that hygrometer calibration will be used for the uncertainty ‘Temperature - Thermometer calibrated correction factor’
• Temperature - Thermometer Rounding - The uncertainty is equivalent to half the measurement increment reading of the instrument used for the calibration. This uncertainty source is assigned 100 degrees of freedom and a coverage factor of √3.
• Certified distances - Uncertainty of Certified Distances - The standard uncertainty of all inter-pillar distances are calculated from the CoFactorMatrix of baseline calibrations. This is done by creating a design matrix for every combination of pillars and calculating another CoFactorMatrix according to the formula:
  $$ Q_2=A_2QA_2^T\ \ \ (Eq. 5.8)$$

  Where:

    Q₂ = The secondary CoFactorMatrix

    Q = The CoFactorMatrix from the LSA of calibrating a baseline

    A₂ = A design matrix for every possible combination of from and to pillars

  The standard uncertainties of all inter-pillar distances are equivalent to the square root of the diagonals of the secondary CoFactorMatrix. These values are stored for each baseline calibration. The inter-pillar distance standard uncertainties that correspond to EDMI calibrations are used to calculate the ‘Uncertainty of Certified Distances’ according to the formula:

    uc_ij = k * S_ij

  Where:

    uc_ij = Expanded uncertainty of a distance between pillar i and pillar j

    k = Coverage factor of a baseline calibration determined by the t-student at 95% and LSA derived degree of freedom (dof) t_(95%,dof)

    S_ij = Standard uncertainty of a distance between pillar i and pillar j

• EDM Calibration - Uncertainty of LSA EDMI Correction - This is a type A uncertainty output by the least squares adjustment used to determine calibration parameters for the calibration of EDM Instrumentation. The uncertainty of the parameters are equivalent to the diagonal elements of the CoFactorMatrix. The ‘Uncertainty of (LSA) EDMI Correction’ is calculated according to the following conditions:

  (a) if the calibration of the EDMI has tested for cyclic errors and found them to be significant:

  $$ S_{IC}=S_{zpc}+S_{scf}D+S_1\sin{\left(2\pi\frac{D}{U}\right)+S_2\cos{\left(2\pi\frac{D}{U}\right)+S_3\sin{\left(4\pi\frac{D}{U}\right)+S_4\cos{\left(4\pi\frac{D}{U}\right)}}}}\ \ \ (Eq. 5.9)$$

  Where:

    S_IC = Standard uncertainty of the Instrument Correction

    D = Distance

    S_zpc = Standard uncertainty of the Zero Point Correction (m)

    S_scf = Standard uncertainty of the Scale Correction Factor (1:x)

    U = Instrument Unit Length (m)

    S₁ = Standard uncertainty of the term 1 - Cyclic (1:x)

    S₂ = Standard uncertainty of the term 2 - Cyclic (1:x)

    S₃ = Standard uncertainty of the term 3 - Cyclic (1:x)

    S₄ = Standard uncertainty of the term 4 - Cyclic (1:x)

  (b) if the calibration of the EDMI has not tested for cyclic errors, has found them to be insignificant:

  $$ S_{IC}=S_{zpc}+S_{scf}D\ \ \ (Eq. 5.10)$$

  The uncertainty is calculated according to the following formula:

  $$ {uc}_{IC}=k * S_{IC}\ \ \ (Eq. 5.11) $$

  Where:

    u_IC = Expanded uncertainty of the Instrument Correction

    k = Coverage factor determined by the t-student at 95% and LSA derived degree of freedom (dof) t_(95%,dof)

Table 5.3 An example set of EDMI measurements observed on a EDM Baseline

Pillars		Raw Distances (m)					Average Distance
From	To	Slope Dist. 1	Slope Dist. 2	Slope Dist. 3	Slope Dist. 4	Slope Dist. 5	Slope Dist. (m)	Dist. Std (mm)
1	2	20.4005	20.4005	20.4006	20.4004		20.40050	0.07
1	3	122.3877	122.3875	122.3875	122.3876		122.38758	0.08
1	4	305.9899	305.9899	305.9900	305.9900		305.98995	0.05
1	5	448.7901	448.7901	448.7901	448.7902		448.79013	0.04
1	6	509.9729	509.9728	509.9728	509.9729		509.97285	0.05
2	1	20.4009	20.4011	20.4009	20.4009		20.40095	0.09
2	3	101.9871	101.9870	101.9869	101.9870		101.98700	0.07
2	4	285.5891	285.5891	285.5888	285.5889		285.58898	0.13
2	5	428.3894	428.3892	428.3893	428.3892		428.38928	0.08
2	6	489.5733	489.5730	489.5732	489.5733	489.5730	489.57316	0.14
3	1	122.3877	122.3877	122.3877	122.3878		122.38772	0.04
3	2	101.9872	101.9870	101.9870	101.9870		101.98705	0.09
3	4	183.6028	183.6027	183.6027	183.6027		183.60273	0.04
3	5	326.4033	326.4034	326.4034	326.4034		326.40338	0.04
3	6	387.5854	387.5858	387.5856	387.5854	387.5855	387.58554	0.15
4	1	305.9901	305.9902	305.9902	305.9903		305.99020	0.07
4	2	285.5894	285.5894	285.5897	285.5895	285.5896	285.58952	0.12
4	3	183.6025	183.6027	183.6026	183.6028	183.6028	183.60268	0.12
4	5	142.8025	142.8026	142.8024	142.8024		142.80248	0.08
4	6	203.9846	203.9846	203.9845	203.9846		203.98458	0.04
5	1	448.7904	448.7904	448.7904	448.7904		448.79040	0.00
5	2	428.3904	428.3903	428.3903	428.3904		428.39035	0.05
5	3	326.4035	326.4035	326.4035	326.4034		326.40348	0.04
5	4	142.8027	142.8027	142.8027	142.8028		142.80272	0.04
5	6	61.1820	61.1819	61.1822	61.1824	61.1823	61.18216	0.19
6	1	509.9726	509.9727	509.9728	509.9727		509.97270	0.07
6	2	489.5721	489.5720	489.5722	489.5724	489.5720	489.57214	0.15
6	3	387.5860	387.5863	387.5865	387.5862	387.5862	387.58624	0.16
6	4	203.9853	203.9851	203.9852	203.9851		203.98518	0.08
6	5	61.1825	61.1827	61.1827	61.1827		61.18265	0.09

Table 5.4 An example set of alignment survey

Surveyed From Pillar	To Pillar	Offset (m)	Average (m)	Standard Deviation (m)
2 3 4 5 6 1	1	0.000 0.000 0.000 0.000 0.000 0.000	0.000	0.001
1 3 4 5 6 2	2	0.112 0.112 0.111 0.113 0.113 0.112	0.112	0.001
1 2 4 5 6 3	3	0.116 0.117 0.117 0.116 0.117 0.116	0.116	0.000
1 2 3 5 6 4	4	0.132 0.131 0.131 0.131 0.132 0.132	0.131	0.001
1 2 3 4 6 5	5	0.151 0.150 0.150 0.151 0.150 0.150	0.150	0.001
1 2 3 4 5 6	6	0.000 0.000 0.000 0.000 0.000 0.000	0.000	0.001

7.1.1 Hypothesis A

This null hypothesis states that the calculated experimental standard deviation, s, is smaller than or equal to the manufacturers specifications value σ.

The null hypothesis is not rejected if the following condition is fulfilled:

Otherwise, the null hypothesis is rejected.

7.1.2 Hypothesis B

The null hypothesis states that the experimental standard deviation, s belong to the same population as the standard deviation (ε̃) obtained in the previous calibrations using the same instrumentation.

The null hypothesis is not rejected if the following condition is fulfilled:

Otherwise, the null hypothesis is rejected.

7.1.3 Hypothesis C

The null hypothesis states that the zero-point correction, δ, is equal to zero as supplied by the manufacturer (δ = 0)? Or, if prisms with a given zero-point correction, δ₀ are used, is δ = δ₀?

The null hypothesis is not rejected if the following condition is fulfilled:

Otherwise, the null hypothesis is rejected.

$$ E_n=\frac{(x_{lab}-x_{ref})}{(U_{lab}^2+U_{ref}^2)}\ \ \ (Eq. 7.4) $$

$$ ΔH_{ij}=(M_{jk}-M_{ik})\ \ \ (Eq. 3.1) $$

$$ v_k=ΔH_{ij}-[M_{jk}-M_{ik} ]\ \ \ (Eq. 3.2) $$

$$ v = AX + w \ \ (Eq. 3.3)$$

A least squares adjustment computes the most probable values for the height differences between the pins. Any changes to the observations should be as small as possible. A least squares adjustment minimises the sum of the squares of the weighted residuals. The observation matrix (Eq. 3.3) can be combined with a weight matrix, $P$ (Eq. 3.5) to derive the best estimate for $\Delta H_{ij}$, represented by $x$ in Eq. 3.3 as follows:

$$x = (A^{T}PA)^{-1} A^{T}Pw \ \ (Eq. 3.4) $$

$$ \sigma = \sqrt{(\sigma_{M_{i}}^2+\sigma_{M_{j}}^2+\sigma_{T_{ij}}^2)}\ \ \ (Eq. 3.5)$$

$ \sigma_{H_{ij}} = \sqrt{({A}^TPA)^{-1}} * 1.96\ \ \ (Eq. 3.6)$

$$ C_f=\frac{ΔH_{ij}}{(M_{j}-M_{i})}\ \ \ (Eq. 4.1) $$

$$ V_{ij}=1 - \frac{ΔH_{ij}}{[M_{j}-M_{i}]}\ \ \ (Eq. 4.2) $$

$$ \left[\begin{matrix} v_1=1-∆H_{1,2}/(M_2-M_1 )\\v_2=1-∆H_{1,3}/(M_3-M_1 )\\\ldots\\v_{(n-1)}=1-∆H_{(1,n)}/(M_n-M_1 )\\\end{matrix}\right]\ \ \ (Eq. 4.3) $$

$$ V=AX+w\ \ \ (Eq. 4.4) $$

To determine the best estimate of $C_f$, the observation matrix (Eq. 4.4) can be combined with a weight matrix, $P$ (Eq. 4.6) as follows:

$$x = (A^{T}PA)^{-1} A^{T}Pw \ \ (Eq. 4.5) $$

$$ P=[ \frac{1}{σ_{k}^2}]\ \ \ (Eq. 4.6)$$

$$ \sigma = \sqrt{(\sigma_{M_{M1}}^2+\sigma_{M_{Mi}}^2+\sigma_{T_{1i}}^2)}\ \ \ (Eq. 4.7)$$

$$ C_{f_{T_0}}=C_f*(1+(\alpha(T_0-T_{obs})))\ \ \ (Eq. 4.9) $$

$$ \Delta H_C=((((T_{obs}-T_0)*\alpha)+1)*C_{f_{T_0}})*\Delta H_{obs}\ \ \ (Eq. 4.10) $$