The rod comparator at the Masala laboratory

4 The description of the field work 20

5.3 The rod comparator at the Masala laboratory

FGI moved to Masala in 1995, and a new rod comparator was constructed [32].

In the new version, an HP 5529A was used as a laser interferometer, and a CCD camera COHU with a Matrox Meteor board was used instead of a measuring microscope. It had an automated weather station with a Vaisala QLI50 interface, HUMICAP MPD35 temperature and humidity sensors, and a PT100 pressure sensor. The rods were moved in a linear rigid conveyer using a stepping motor and the movement was balanced with counterweights. The comparator was controlled by Visual Basic controlling software.

In rod scale calibrations, the positions of the graduation lines were measured twice from the bottom to the top and the back at three temperatures. One cali-bration lasted about 90 min depending on the type of rod scale. The measuring accuracy of the calibration depended on the quality of the rod scale, and with 95% confidence it was between 0.7 ppm and 2.0 ppm [33]. The thermal expan-sion coefficient was based on the measurements of one graduation line interval at different temperatures. The accuracy, which was obtained from six independent measurements was approximately 0.2 (µm/m)/

^◦

C. In 2002 system calibration was added to the measuring features [34]. In

sys-tem calibrations rod readings from the levelling instrument are compared to the

laser interferometer readings and thus the rod corrections include not only rod

scale information but also how well instruments interpret the scale [35].

Sys-tem calibration corrections were not utilized in the rod corrections of the Third

Levelling observations.

Chapter 6 The computation of the N2000 heights

In this chapter the computation of the Finnish levelling observations and the adjustments are presented. The selection of the EVRF2000 datum was originally based on the work done in the Working Group for Height Determination of the Nordic Geodetic Commission (NKG). Before the adjustments, the observations were corrected to the system epoch 2000.0 using the Nordic land uplift model NKG2005LU.

In 2002 the General Assembly of the Nordic Geodetic Commission (NKG) ac-cepted a resolution, which considered it desirable that the Nordic countries “adopt [height] systems with minimal differences from each other and from the European Vertical Datum”. Following the NKG proposal [36] the Technical Working Group (TWG) of the International Association of Geodesy (IAG) and its subcommission for Europe (EUREF) recommended a close co-operation between the NKG, all countries around the Baltic, the Netherlands, and the United European Levelling Network (UELN) computing centre. Subsequently, Estonia, Latvia, Lithuania, Poland, Germany and the Netherlands made their levelling data used in the EVRF2000 available to the NKG.

The N2000 height system differs only a little from its Nordic counterparts due to the joint BLR adjustment and the inclusion of levelling lines from neighbouring countries. Additionally, the new Swedish height system RH2000 is based on the adjustment of the BLR data [37]. The difference between the Finnish and Swedish height systems is less than 2 mm at the boundary zone [38]. Both height systems are based on the adjustment of the BLR data, so the countries have the same datum, land uplift model, and weighting of observations. Comparison with the European Vertical Reference Frame 2007 (EVRF2007) [39] shows that the N2000 heights are about 9 mm greater than the EVRF2007 heights.

In the first adjustment step, the height of the N2000 main point PP2000 was computed using the collected data. In the second step this value was fixed in the N2000 adjustment. The fundamental bench mark PP2000 is in Kirkkonummi at the Mets¨ ahovi Research Station.

Rod readings, sight distances, temperatures and other collected data were

combined and associated corrections calculated before the adjustments. The

pre-adjustment reductions were presented in the line papers i.e. the observation

documents of entire lines, which were computed annually after the field seasons.

At that stage, the height differences were in their observation epochs.

6.1 Corrections

The applied corrections are: refraction, rod scale, tidal deformation, and in the case of the Zeiss Ni002 level, the influence of the Earth’s magnetic field. The corrected metric height difference ∆H m is:

∆H m = ∆ H m,obs + C ref + C rod + C tidal + C magn + C tidal,p (6.1) where:

• C ref is the refraction correction due to vertical air temperature differences,

• C rod is the rod correction which takes into account the change in rod scale length in varying air temperatures,

• C tidal is the tidal correction for the crustal deformation during the mea-surement due to tidal deformation of the Earth,

• C magn is the magnetic correction for the Zeiss Ni 002,

• C tidal,p is the permanent tidal deformation.

The metric height differences were converted into geopotential differences us-ing the mean gravity of the bench mark interval (Formula 6.2). The gravity related height difference in geopotential units is:

∆H

_gpu

= 0.5(g

₁

+ g

₂

)∆H

(6.2) where:

• ∆H

gpu

is the geopotential difference, (10 m

s

⁻²

),

• g

₁

and g

₂

are the interpolated gravity values at bench marks 1 and 2, (10 ms

⁻²

),

• ∆H

is the metric height difference, (m).

The geopotential difference is about 2% smaller than the corresponding metric difference. A height difference of one metre is about 0.98 gpu or 980 mgpu. The gravity values were interpolated from the five kilometre gravity grid of the First Order Gravity Network of Finland [40].

6.1.1 Refraction correction

By definition, the levelling refraction is the bending of the sight line from the horizontal level due to changes in the refractive index along the path of the line-of-sight. The correction is based on the works of Kukkam¨ aki [41, 42]. The refraction correction in mm for one setup using the Kukkam¨ aki formula is

C ref = −10

⁻⁵

· 70 · s

Figure 6.1: The land uplift model NKG2005LU and the network of the Baltic

Levelling Ring. The black lines belong to the network of the N2000

adjust-ment. Isobases show the vertical velocity in mm/yr relative to the mean sea

level (1892–1991). Outside the –2 mm/y isobase the value is set to be constant

–2 mm/year. The red dots indicate fixed bench marks in the BLR and N2000

adjustments.

• The constant of 70 was proposed by Hyt¨ onen[24]. Determination of the parameter is based on a vertical temperature distribution and the height of a levelling instrument [41, 42, 43],

• s is the sight distance (m),

• ∆T is the temperature difference above the ground: T(2.5 m)-T(0.5 m),

^◦

C

• ∆H is the height difference in mm.

6.1.2 Taavitsainen formula for the temperature gradient estimation

The Taavitsainen prediction model [25] was used in the estimation of missing temperature differences. The input data for the model includes a zenith distance of the Sun z (degrees); short-period shimmering (turbulence) of air v, expressed in whole numbers from 0 to 3; air temperature T (

^◦

C); cloud cover c, expressed in whole numbers from 0 to 10 and wind speed w (m/s). Only the Sun’s zenith distances are precise values. The surveyors estimated the input values for v, c and w.

If the temperature data was missing due to rain, then the constant value of -0.1

^◦

C was used. Consequently, the Taavitsainen predictions were not used for rainy day observations. If the input data for the Taavitsainen model was not complete, then the value of -0.2

^◦

C was used.

The predicted values were corrected using ground surface information. On dirt roads the temperature differences are smaller than above railway or asphalt surfaces. Taavitsainen predictions were accepted on asphalt roads and on crushed stone along railways. If a ground surface was only partly covered with asphalt or railway crush stone, the predicted value was multiplied by 0.75. If a levelling route was on unpaved roads, then only a half of the predicted value was used.

The Taavitsainen formulas were different for the spring and autumn seasons and for the morning and evening hours. Formulas 6.4 and 6.6 were used if the observations were performed before noon. With the afternoon and evening ob-servations, Formulas 6.5 and 6.7 were used. The spring season formulas are:

∆T = 0.74772 − 0.03961z − 0.21019v − 0.016054c − 0.05993w + (6.4) +0.00403zc + 0.00148T

+ 0.00849c

∆T = −0.38037 − 0.00170zT + 0.00103zc − 0.02322vc + (6.5) +0.00140T

+ 0.00133T c

The autumn season formulas are:

∆T = 0.361 − 0.015z − 0.020c − 0.17vT − 0.004T w + 0.007cw, (6.6)

∆T = 1.161 − 0.025z − 0.073T − 0.182c + 0.002zc − 0.053vw + (6.7)

+0.002T

+ 0.011c

+ 0.002w

6.1.3 Rod correction

Rods were calibrated before and after field seasons. The rod scale lengths were assumed to change linearly between the calibration epochs. A thermal expansion coefficient was the average value from the calibrations. For every rod pair, one model was used for the spring season (January–June) and another model was used for the rest of the year. The calibrations are presented in Appendix B. The rod correction in µm is:

C rod = ( λ + α (T − 20

^◦

C)) ∆H (6.8) where

• λ is the rod scale correction (µm/m) at 20

^◦

C,

• α is the expansion coefficient (µm/m)/

^◦

C,

• T is the temperature (

^◦

C) and

• ∆H is the height difference (m).

6.1.4 Tidal correction

The Earth’s temporal tidal deformation is corrected using the formulas and com-puter programs by Heikkinen[44].

For the Earth’s permanent tidal deformation, a mean tidal system and a zero tidal system are used in the computations. In a zero tidal system, the permanent tidal attraction of the Moon and the Sun is removed, but the resultant permanent tidal deformation of the Earth is retained. In a mean tidal system the permanent tidal deformation and the tidal attraction is retained. In the previous height systems, the permanent tidal deformation was in a mean tidal system (mean geoid), which represents the natural behaviour of the Earth’s crust and is approximately the mean sea level.

In the N2000 height system, the permanent tidal deformation is in a zero tidal system, but the Finnish and other European levelling observations were computed and adjusted in a mean tidal system. The difference between the mean and zero tidally corrected heights Ctidal,p is computed relative to NAP using Formula 6.9 [45]:

C tidal,p = ∆H m-z = H mean − H zero (6.9)

= 29.6 sin

ϕ − sin

ϕ

NAP

cm where

• H mean is the height in a mean tidal system,

• H zero is the height in a zero tidal system,

• ϕ

_NAP

= 52.38137

^◦

is the latitude of NAP,

• ϕ is the bench mark’s latitude.

Table 6.1: Magnetic field coefficients for the Zeiss Ni002 levelling instruments

In the observation list ∆H m-z is presented in geopotential units. The trans-formation from cm to mgpu is done using the normal gravity [46]

γ

= 978032.67715(1 + 0.0052790414sin

ϕ + 0.0000232718sin

⁴

ϕ +0.0000001262sin

⁶

ϕ + 0.0000000007sin

⁸

ϕ)10

⁻⁵

m/s

(6.10) This is the normal gravity on the surface of the GRS80 reference ellipsoid. In Finland, the value is from 9.819 m/s

to 9.826 m/s

.

6.1.5 Magnetic correction

The magnetic corrections are applied to the Zeiss Ni002 observations. Rumpf and Meurisch presented the point that automatic levels are sensitive to the Earth’s magnetic field[47]. At the Finnish Geodetic Institute, Kukkam¨ aki and Lehmuskoski studied this phenomenon [16] by placing instruments into a Helmholtz coil, which generates a strong magnetic field. By repeating observations on differ-ent magnetic field strengths, they estimated the influence of the magnetic field on the levelling instruments. The magnetic field coefficients are presented in Table 6.1.

The influence of the Earth’s magnetic field is corrected by the formula:

C magn = H

15000 M · S · cos(A + D) (6.11) where

• H

is the horizontal intensity of the magnetic field (nT),

• M is the magnetic field coefficient of the instrument (mm/km),

• S (km) is the length of the straight line between the bench marks.

• A is the azimuth of the bench mark interval and

• D is the declination of the magnetic field,

Properties of the magnetic field were extracted from the magnetic charts [48].

In the levelling computations the declination of the magnetic field has been com-puted using the formula:

D = (27.6 − 0.66∆ϕ + 33.97∆λ − 0.291(∆ϕ)

− (6.12)

−0.185(∆λ)

+ 1.163(∆ϕ∆λ)/60

and the horizontal intensity using the Formula:

H

= 13879.6 − 412.43∆ϕ + 29.11∆λ − 1.367(∆ϕ)

− (6.13)

−2.259(∆λ)

+ 0.802∆ϕ∆λ

where the latitude difference ∆ϕ is ϕ − 63

^◦

and the longitude difference ∆λ is λ − 16

^◦

.

6.1.6 Land uplift correction

The recommendation of the NKG was followed to correct the height differences to epoch 2000.0 with the land uplift model NKG2005LU (Figure 6.1). This model is a combination of the geophysical land uplift model by Lambeck et al.[49] and Vestøl’s empirical land uplift model [50]. The description of the NKG2005LU model is presented in [37]. This land uplift model was also used with the new height system adjustments in Sweden and Norway, and later with the European Levelling network adjustment EVRF2007 made by the United European Levelling Network (UELN) computing centre [39].

Lambeck’s model covers the whole area of the Baltic Levelling Ring. It em-ploys tide gauge observations mainly at the Baltic Sea and information about the tilting of the water level of the largest lakes in Sweden and Finland.

Vestøl’s empirical land uplift model is based on the repeated precise levellings, uplift rates from the continuously operating GPS stations [51], and the tide gauge uplift rates for 58 tide gauges around the Baltic and adjacent waters [52]. The data in the model includes the three Finnish precise levellings. One disadvantage is that Vestøl’s model does not cover the whole area of the Baltic Levelling Ring.

The NKG2005LU uplift rates (mm/y) were converted into mgpu/y by multiplying it with normal gravity γ

₀

(Formula 6.10).

The land uplift correction is computed using the difference between the ob-servation and system epochs and the uplift rate difference:

C upl = (2000 .0 − t)(L end − L start) (6.14) The geopotential difference at the system epoch 2000.0 is:

∆H mgpu,2000 = ∆ H mgpu,t + C upl . (6.15) In these formulas:

• t is the observation epoch,

• ∆H mgpu,t is the observed geopotential difference, and

• L start and L end are the land uplift rates of the start and end bench marks (mgpu/y).

6.1.7 Example. Corrections

As a computation example is a bench mark interval 35007-78016, its height

dif-ference in the zero tidal system is -80.47 mgpu. The land uplift values of the

bench marks 35007 and 78016 are 2.303 mgpu/y and 2.318 mgpu/y, respectively.

If the observation epoch is 1979.75, then the land uplift correction (Formula 6.14) for the observation is:

C upl = (2000.0 − 1979.75) y · (2.318 − 2.303) mgpu/y

= 20.25 y · 0.015 mgpu/y = 0.31 mgpu.

If the height difference in the observation epoch is -80.47 mgpu, then the land uplift corrected value in the epoch of 2000.0 would be:

−80.47 mgpu + 0.31 mgpu = −80.16 mgpu.

In the observation list, the height differences are in the zero tidal system.

From Formula 6.9 it easily follows that the height difference in the mean tidal system from start to end would be:

∆H mean = ∆ H zero + C tidal,zero

→

mean , (6.16) where

C tidal,zero

→

mean = ∆ H m-z,end − ∆H m-z,start mgpu . (6.17) The tidal system differences H mean −H zero are 36.52 mgpu (BM 35007) and 36.56 mgpu (BM 78016). Therefore, the correction is: