Vertical reference systems based on national precise levelling networks have been realised in Europe since the 1860’s. The heights are related to gravity, which by convention may be measured values or amodel. The Norwegian Height System was adjusted in 1956 when the southern part of the country had been levelled once. Because of the lack of measured gravity, Clairaut’s formula for gravity was used in the orthometric correction. The intention was to establish an orthometric height system, but this was never tested or confirmed. In the literature the system is often referred to as orthometric, but some times as normal orthometric. This study shows that the Norwegian Height System was realised by a strongly deformed net due to lack of land uplift corrections. The derived heights are neither normal nor orthometric. When the land uplift is handled in a proper way, the heights are close to normal heights. The remaining small differences are shown to depend on the free air anomaly

Lysaker, Dagny Iren

Mathisen, O.

Pettersen, B.R.

Solheim, Dag

English

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Nordic Journal of Surveying and Real Estate Research Volume 3, Number 1, 2006ordic Journal of Surveying and Real Estate Research Volu e 3, u ber 1, 2006Nordic Journal of Surveying and Real Estate Research 3:1 (2006) 7–19submitted on 23 September 2005revised on 27 April 2006accepted on 16 May 2006The Norwegian Height System NN1954 RevisitedD. I. Lysaker, B. R. Pettersen, O. MathisenDept. of Mathematical Sciences and Technology, Norwegian University of LifeSciences, P. O. Box 5003, N-1432 Ås, Norway; dagny.lysaker@umb.noD. SolheimGeodetic Institute, NMA, Serviceboks 15, N-3504 Hønefoss, Norway;dag.solheim@statkart.noAbstract. Vertical reference systems based on national precise levelling net-works have been realised in Europe since the 1860’s. The heights are relatedto gravity, which by convention may be measured values or a model. TheNorwegian Height System was adjusted in 1956 when the southern part ofthe country had been levelled once. Because of the lack of measured gravity,Clairaut’s formula for gravity was used in the orthometric correction. Theintention was to establish an orthometric height system, but this was nevertested or confirmed. In the literature the system is often referred to as or-thometric, but sometimes as normal orthometric. This study shows that theNorwegian Height System was realised by a strongly deformed net due tolack of land uplift corrections. The derived heights are neither normal nororthometric. When the land uplift is handled in a proper way, the heightsare close to normal heights. The remaining small differences are shown todepend on the free air anomaly.Key words. NN1954, normal height, spheroidal-orthometric height, ortho-metric height, postglacial rebound1 IntroductionEarly attempts to establish a height system for Norway were rooted in the 1864general assembly of the Mittel-Europäische Gradmessung in Berlin. A tide gaugewas mounted in the harbour of Oslo in 1876 to provide a reference level by averag-ing sea level recordings over several years. Levelling began in 1877, but progresswas slow. In 1890 a reference marker for levelling was established on the propertyof the Geographical Survey of Norway in Oslo and its position relative to the tidegauge was determined.In 1916 modern levelling instruments were acquired and the observing pro-gram gained momentum. Measurements were made along primary communica-tion lines south of the Arctic Circle, but the calendar would show 1953 before the8 The Norwegian Height System NN1954 Revisitede r e i ei t Ryste evisitesouthern part of the country (between latitude 58.0◦and 66.3◦) was covered. In themeantime, tide gauges had been set up at several locations along the coast. Thetime series revealed a long term slope due to postglacial uplift, but the slope valuevaried from site to site. In Oslo it was found to be about 3 mm per year. Oslowas thus considered an unpractical site for a reference marker, and a new one wasestablished at Tregde (near the south tip of Norway) where the measured uplift isvery close to zero. The actual determination of apparent mean sea level was basedon an average of seven tide gauges along the coast.The entire data set was adjusted in 1956 and the result reported in (Trovaagand Jelstrup, 1956). The height system is called NN1954. A fundamental problemarises because the observing program needed almost 40 years to be completed in aregion where postglacial rebound effects had been known to exist for a long time.No reliable model existed, however, so the adjustment procedure did not take landuplift into account at all. An additional problem was caused by the lack of observedgravity values along the levelling lines and loops. Thus Clairaut’s formulaγClairaut = γ0(1−αcos2φ−βh) (1)was used to obtain normal gravity values for a chosen reference Earth spheroid.Here is α= 2.6444×10−3, β= 0.314×10−6m−1, h is in meters and γ0 is normalgravity for latitude 45◦. The gravity difference between two nearby points at equalheight is obtained by differentiating Clairaut’s formula,∆γClairaut = γ02αsin 2φ∆φ, (2)which is valid when ∆φ is small. Referring to Fig. 1, the geopotential numbers atA and B are interrelated byHBḡ2 = HAḡ1 +(∑∆n)gm, (3)where ḡ1 and ḡ2 are mean gravity values along the plumblines of A and B, respec-tively. For short distances AB the gravity for the whole line can be approximatedby the average of the end points,gm =ḡ1 + ḡ22Introducing ∆g = ḡ2 − ḡ1, we obtainHB = HA +∑∆n− ∆gḡ2�HA +∑∆n2�. (4)The parenthesis represents the average height, h, between A and B and thelast term may be written as ∆gḡ2 h. Since no observed gravity values were available,Nordic Journal of Surveying and Real Estate Research Volume 3, Number 1, 2006r i r l f r i l st t s r l , r ,Figure 1. The relationship between the orthometric heights and gravity (i.e. geopotentialnumbers) in two points on a levelling line.they were replaced by the corresponding normal values. Arguing that this last termis small, ḡ2 was replaced by γ0 to obtain a spheroidal-orthometric correction ofcorr =∆γClairautγ0h = 2αsin 2φ∆φh. (5)The levelling net north of the Arctic Circle was originally referred to the tidegauge at Narvik and was thus a separate entity. It was combined with the net southof the Arctic Circle in 1974 to generate a common height system for mainlandNorway. The analysis in this paper considers the region south of the Arctic Circleexclusively, and does not address the attachment and combination of the northernbranch.2 Data and MethodologyThis study aims to determine whether the original realisation of the NorwegianHeight System yields normal heights or orthometric heights. This is done bycomparing the official height system with theoretically “ideal” calculated normalheights and orthometric heights.2.1 HeightsPrecise height determination in wide areas must apply geopotential numbers, sincelevelling alone does not yield unambiguous height values. Levelling is combined10 The Norwegian Height System NN1954 Revisitedr i i t Ryste isitwith gravity, and heights are derived from the formula, (Heiskanen and Moritz,1967, page 171):height =CG, (6)where C is geopotential number and G is gravity. The type of height obtaineddepends on the choice of gravity G. If mean gravity along the plumb line is used,orthometric heights are achieved. The use of mean normal gravity γ̄, however,yields normal heights. In some regions, such as Norway, postglacial rebound mustbe considered as well in order to achieve accurate height values.Historically, geopotential numbers were not used directly. Corrections de-rived from geopotential numbers were applied to the levelling values instead, (Heis-kanen and Moritz, 1967, chap. 4). Unambiguous heights were achieved by,height2 = height1 +∆n+ corr. (7)This last approach was used when NN1954 was established. The corrections werecomputed according to Eq. (5). The type of height depends on the type of correc-tion.In the present investigations “ideal” normal heights and orthometric heightsare obtained by adjusting geopotential numbers corrected for land uplift, and bycalculation of mean normal gravity values γ̄, and mean gravity values ḡ along theplumb line, respectively. The mean normal gravity is given by:γ̄= γ�1− (1+ f +m−2 f sin2φ)H∗a+H∗2a2�(8)where f is the flattening, a is the semi major axis of the GRS80 ellipsoid, γ is normalgravity at the ellipsoid, m is the Clairaut’s constant, H ∗ is the normal height and φis the geodetic latitude, (Heiskanen and Moritz, 1967, p. 170). The mean gravityis approximated by:ḡ = g+0.0424H (9)where g is measured gravity at the Earth’s surface and H is orthometric height. Thereal mean gravity cannot be found since we do not know the density of the Earth,and cannot measure gravity inside the Earth’s crust. (More details in (Heiskanenand Moritz, 1967).) Both mean normal gravity and mean gravity are calculated inan iterative process, since they depend on the height.2.2 The official height system databaseThe official height database of the Norwegian Mapping Authority contains heightvalues and coordinates in the EUREF89 frame for all geodetic markers ever in-cluded in precise levelling in Norway. All original levelling measurements since1916 are listed, with corresponding spheroidal-orthometric corrections and gravityvalues for each marker, either measured or computed from existing nearby gravityNordic Journal of Surveying and Real Estate Research Volume 3, Number 1, 2006r i r l f r i l st t s r l e , u ber 1, 2006observations. An algorithm allows geopotential numbers and height differencesbetween any two markers to be generated, with or without spheroidal-orthometriccorrections and land uplift corrections according to a model by (Danielsen, 2001).2.3 The uplift model of DanielsenThe land uplift model of (Danielsen, 2001) covers Fennoscandia. It is based onfirst order levelling made by the Finnish Geodetic Institute, the National LandSurvey in Sweden, the Norwegian Mapping Authority (SK) and the NorwegianState Railways, repeated GPS measurements and many of the mareographs in thearea. Two methods, least squares collocation and “geometric filter”, are used todetermine the uplift rates. They produce separately images of the land uplift whichare almost identical, and with quite similar accuracy. The model used here is basedon least squares collocation. The formal accuracy is about 0.5 mm/year and variesslightly throughout the country (Danielsen, 2001).3 Calculation and results3.1 The Norwegian Height SystemIn order to check our adjustment procedure we first reconstructed the official heightsystem using the original height differences with spheroidal-orthometric correc-tions added. The overall agreement was better than 3 mm.Next we adjusted a set of geopotential numbers for which the empirical upliftmodel of Danielsen had been applied. The geopotential numbers are thus referredto the same epoch, 1954. The reference marker at Tregde is kept fixed, as in theoriginal adjustment. “Ideal” orthometric heights and “ideal” normal heights werethen calculated as described above, Eq. (6). The differences relative to the officialheights are shown in Fig. 2 and 3.Fig. 2 and 3 show small differences in the south, but generally the differencesare large. Most markers differ by more than 8 cm. In Fig. 2 the differences appearto increase with altitude and also in general towards north-east. This latter effectmay also be noted in Fig. 3, but to a lesser extent. The most noticeable differencesare however, located along a section of the Norwegian-Swedish border. In the sameregion the uplift model of Danielsen shows the largest uplift values. NN1954 is astrongly deformed net, and it is impossible to conclude whether the official heightsare orthometric heights or normal heights.The date of observation explains some of the deformation. The first levellingreached the border in the 1930’s. An uplift of 4–5 mm/year in the area betweenlatitude 60◦ and 63◦ produces a mismatch of approximately 9 cm in 2 decades(1954–1930). Thus the differences are 9 cm too large here. This is especially truenear Røros. Observations from 1918 and an uplift of 4–5 mm/year produces anuplift correction of 15 cm (1954–1918).More recent measurements have not been corrected for land uplift either. Theerrors caused by the omission of the uplift corrections in the 1956 adjustment prop-12 The Norwegian Height System NN1954 Revisited12 The Norwegian Height Rystem NN1954 RevisitedFigure 2. Orthometric heights minus NN1954 heights.agates into the expansions of the net. Some levelling lines are pure expansions, andcorrections to these lead to quite small differences. The levelling line located in-side circle A in Fig. 3 and 5 is an example of this. This line was measured in 1983,and with a land uplift of approximately 3.5 mm/year the correction would be 10cm if we refer back to 1954. The differences in Fig. 3 are for this line between10 cm and 12 cm. When the uplift effect is corrected for in the height system, thedifferences are reduced to 0–2 cm (see Fig. 5).Nordic Journal of Surveying and Real Estate Research Volume 3, Number 1, 2006r i r l f r i l st t s r l , r ,Figure 3. Normal heights minus NN1954 heights.Other levelling lines are forced into the existing net and a correction herewould only take away some part of the deformation. The levelling line insidecircle B in Fig. 3 and 5 is such a line. In the north the line is attached to a pointmeasured in 1918, and in the south to a point measured for the first time in 1935.14 The Norwegian Height System NN1954 RevisitedThe or e i ei t Ryste NN1954 RevisitedFigure 4. Orthometric heights minus NN1954 heights corrected for land uplift.3.2 Corrections for land upliftThe effect of postglacial rebound on the NN1954-data is significant. The databaseentries are thus corrected by applying the empirical uplift model of (Danielsen,2001). Only then it is hoped to decide if the use of Clairaut’s gravity insteadof measured gravity yields normal heights or orthometric heights. Therefore theofficial height system was recalculated by referring to one joint epoch, 1954.A third dataset from the database was used for this purpose. Spheroidal-Nordic Journal of Surveying and Real Estate Research Volume 3, Number 1, 2006ordic Journal of Surveying and eal state esearch olu e 3, u ber 1, 2006Figure 5. Normal heights minus NN1954 heights corrected for land uplift.orthometric corrections and the empirical uplift model of (Danielsen, 2001) wereapplied to the height differences. The set was adjusted and compared with “ideal”orthometric heights and “ideal” normal heights. The results are shown in Fig. 4and 5.Now the large differences along the Swedish border diminish in both figures.This demonstrates that almost all deformation shown in Fig. 3 and 2 was due16 The Norwegian Height System NN1954 RevisitedThe or e i ei t Ryste NN1954 RevisitedFigure 6. Orthometric heights minus NN1954 heights corrected for land uplift, plottedagainst height. A second order polynomial has been fitted to the data.to the postglacial rebound. In Fig. 4 the differences increase with altitude. TheNN1954 heights corrected for uplift are clearly too small in the mountainous areascompared to orthometric heights.(Heiskanen and Moritz, 1967) chapter 8–13 expresses the difference betweennormal heights and orthometric heights by the height and the Bouguer anomaly.H∗−H ≈∆gBγ̄H. (10)The Bouguer anomaly, ∆gB, depends on the height as well, so the differencebetween normal heights and orthometric heights should vary quadratically withheight. This is confirmed to hold for the Norwegian Height System (cf. Fig. 6).By fitting a quadratic function we obtain a correlation of 0.9.Fig. 5 shows differences in the interval -2–3 cm, but most points have differ-ences less than 1 cm. The largest difference in Fig. 3 is 17 cm and the difference inthat point reduces to 0.6 cm in Fig. 5 when uplift is corrected for. The mean differ-ence reduces from 8 cm to 0.7 cm and the standard deviation from 4 cm to 0.8 cm.NN1954 heights corrected for land uplift in meterDifference in cmNordic Journal of Surveying and Real Estate Research Volume 3, Number 1, 2006r i r l f r i l st t s r olu e , Number 1, 2006Figure 7. Height difference’s (i.e. normal heights minus NN1954 heights corrected forland uplift) dependency on free air anomaly.Applying the uplift model of (Danielsen, 2001) really improves the height systemand yields a more homogeneous net. The quadratic dependence and the small dif-ferences in Fig. 5 argues that the use of Clairaut’s computed gravity yields heightsclosest to normal heights.There are some small differences remaining in Fig. 5. The only distinctionbetween normal heights and NN1954 heights corrected for land uplift is the typeof gravity used in the computation. Measured gravity values are used along thelevelling lines to obtain normal heights. The uplift corrected NN1954 heights,however, use Clairaut’s gravity. Clairaut’s gravity and normal gravity differ by no18 The Norwegian Height System NN1954 Revisited18 he or egian eight Ryste 1954 evisitedmore than 2 mGal within the test area of this study. Eq. (1) is thus an early formulato compute normal gravity at a certain altitude. The difference between measuredgravity and normal gravity is the free air anomaly. The variation of the remainingheight differences in Fig. 5 thus depends on the free air anomaly.Fig. 7 shows this dependency. The free air anomaly (red line) is negative inthis case, thus measured gravity is less than the normal gravity. If the levelling lineis followed not much happens with the height differences (green line) as long asthe terrain (blue line) is not changing much. Once the terrain starts to climb grav-ity decreases. The normal gravity decreases more than measured gravity becauseof the attraction of the topographic masses. Thus the free air anomaly increases.The normal heights lose relative to NN1954 heights corrected for land uplift andcontinue to lose as long as the altitude increases. When the terrain descends, theopposite happens. The free air anomaly decreases because normal gravity de-creases more than the measured gravity. Normal heights gain relative to NN1954heights corrected for land uplift and the height difference increases.4 ConclusionThe Norwegian Height System yields neither orthometric nor normal heights. Itis a strongly deformed net which should be recalculated in order to yield reliableheight values. Compared to orthometric heights, NN1954-heights are generallytoo low in the mountainous regions. The fit is better in the lowland, but thereare some areas with differences much larger than average. NN1954 heights arethroughout the country closer to normal heights, especially in the mountains, butlarge differences appear here as well.When the net is corrected for the land uplift, the heights remain clearly toolow in the mountainous areas compared to orthometric heights, while they fit wellto normal heights over the whole region. Fig. 4 is a typical picture of differencesbetween orthometric heights and normal heights. Since we are comparing therecalculated net with orthometric heights this reveals the recalculated heights to beapproximately normal heights. The quadratic dependence in Fig. 6 does also showthis.It is clear that the method used when NN1954 was established yields re-sults very close to normal heights, (provided small land uplift and modest free airanomalies). The remaining differences are due to the difference between observedgravity and normal gravity, the free air anomaly.Acknowledgements. It is a pleasure to thank Olav Vestøl for providing access to theoriginal precise levelling data for Norway.Nordic Journal of Surveying and Real Estate Research Volume 3, Number 1, 2006r i r l f i olu e , Number 1, 2006ReferencesDanielsen, J. (2001). A land uplift model of Fennoscandia. Survey Review, (36):282–291.Heiskanen, W. A. and Moritz, H. (1967). Physical Geodesy. Institute of Physical Geodesy,Technical University, Graz, Austria.Trovaag, O. and Jelstrup, G. (1956). Precise levelling in South Norway. Geodetic publi-cations No. 5, Geographical Survey of Norway.

The Norwegian Height System NN1954 Revisited

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The Norwegian Height System NN1954 Revisited

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