What kind of levels are NN1954?

Transcription

1 What kind of levels are NN1954? Geodesy and hydrology days, Hønefoss November 2003 Dagny I. Lysaker English translation: Paweł Miłosz Szubtarski Table of contents 1 Introduction Generally about heights Non uniqueness (uncertainty) of leveling Orthometric height Normal Heights Height Determination Corrections NN Cohesion (equalization) in NN1954 today Comparison Reconstruction of NN Ideal heights The difference due to real orthometric heights The difference to normal levels Deformation of the grid NN1954 corrected for Post-glacial Rebound Conclusion Bibliography... 18

2 1 Introduction The concept of height has no meaning without a reference surface. Height is the distance from a reference surface to the place we are looking for height. Reference surface can be a physical surface or a mathematical surface. In geodesy, we use several reference surfaces and can therefore speak about more types of height, eg. orthometric heights and normal heights. Precise height determination in larger areas should be made using the geopotential, since Leveling alone do not provide clear heights. Leveling is combined with gravity measurements and then balanced geopotential. Heights are derived from the simple formula: where C the geopotential and G is gravity. What type of height we get depends on what kind of weight is used. It may be slightly inaccurate to say that the measured weights give orthometric heights and weights are theoretical for normal heights. An alternative to cancel out the geopotential would be to inflict a correction of the values of leveling so that the unique levels can be found directly. When NN1954 was equalized, there was an intention to form a network with orthometric heights, so that one used a form of orthometric correction which measured gravity is included. But the weight was missing, so instead of measuring weights, a theoretical gravity is used as a substitute. It has therefore recently been realized that NN1954-heights are closer to normal levels than orthometric heights. Today we can base on measured weights generate weight values in a given position. We thus have the opportunity to calculate the true orthometric heights and normal heights for all leveling permanent marks. A comparison will be able to say what kind of heights NN1954 is. 2 Generally about heights Height is defined in the standard Norway's official height system and reference levels as: ''A point vertical distance of a physical or mathematical reference surface.'' Geoid is such a physical reference surface. It can be defined as the equipotential surface of the Earth's gravity field that best fits the mean sea level at a given epoch''. Heights referring to geoid are therefore virtually coincide with the general perception of height concept in Norway, namely the number of meters above sea level. 2.1 Non uniqueness (uncertainty) of leveling When we use a potential surface as reference surface this creates certain problems. Because of earth s shape the potential surfaces are not parallel with gravity field. This means that leveling alone does not provide unequivocal heights when leveling large areas. The figure below can help to understand: Page 2

3 Figure 2-1: Nonparallel potential surfaces We have a water mirror at rest, but the orthometric height is different at the ends of the water. A leveling from point A to B and along the lake to C gives a height. However, if we are going from D to C, we will have a different height. Both of these heights again may differ from orthometric height. In the figure, a steady stream of water at the C run uphill to B. From physics, we remember that work is equal to force times way: The effort to raise a lot from one potential surface to another in a conservative force field is independent of the way we go. This we exploit by combining leveling with gravity measurements. We look at potential differences instead of pure height differences. Potential difference between a point and geoid called geopotential number or just geopotential. Geopotential to the point is: Regardless of the way we go from P 0 to P, the P will have the same geopotential. As we know the geopotential, the height can be derived. 2.2 Orthometric height Orthometric height, H is the length of the curved vertical line from a point down to geoid. Geopotential in point is uniquely given by: where it is integrated along the vertical line. This includes orthometric height and we can get it out by transforming the formula above. We derive: where Page 3

4 is the average weight along the vertical line. Orthometric height is thus given by: if we know the mean weight along the vertical line. Mean weight can be written as where g(z) is the real weight of an arbitrary point on the vertical line of height z. Now we have a definition of orthometric height, but still do not know the mean weight along the vertical line. This cannot be measured, then we cannot measure in earth. It can still be calculated with sufficient accuracy by setting g(z)=g P +0,0848(H-z), where g P is the weight measured on the soil surface. Mean weight in the soil is then: 2.3 Normal Heights We imagine that the Earth is a spheroid with uniform mass distribution. This land has topography, but the mountains are a lot of resolve. We put a gravitational field around this ellipsoid Earth, a normal gravitational field where the normal potential is given when the position is given. Normal height is then distance along the curved vertical line, but now from reference ellipsoid and up to a point Q where the normal potential U Q equals real potential in the point W P, see Figure 2-2. Figure 2-2: Normal Height The practical usefulness of normal levels may seem a little remote as geoid height is great. We get the height of a point somewhere inside the earth. Aside, we measure however the height from P to Page 4

5 down and the normal heights even define a reference surface. There is this thing called quasigeoid, and it turns out that quasigeoid deviates very little from geoid. Just as we derived an expression for orthometric height, we can derive an expression of normal height. The difference is that we have given an expression of normal weight and can find the mean normal weight precisely by the following formula: Formula 2.2 where m is a help-quantity, f is reference ellipse flattening, a is the ellipse longer radius and φ is latitude. Normal height is given by: The advantage of normal levels is that we avoid the problem that we do not know the weight inside the Earth. 2.4 Height Determination Precise height determination in larger areas should therefore be done by leveling out geopotential and then derived heights from the formula orthometric heights normal heights As an alternative, we can add a correction on leveling values so that they provide unique levels directly. We can derive different corrections depending on the height type we want. 2.5 Corrections We look more closely at orthometric correction. It can be deduced from the geopotential. We get: We add and subtract an arbitrary value to each part, then we divide every part on this value, for example standard normal gravity at 45 degrees latitude. Page 5

6 We order a bit and get orthometric correction equal to: The method of correcting values of leveling was the most commonly used as our height was evened out (equalized). In the orthometric correction are included measured weight values in all aspects, but it lacked the cohesion of the Norwegian height net. Instead there were held Clairauts formula for gravity: where is standard normal gravity at 45latitude. α and β are given constants and h is mean height. Applying Clairauts formula on the expression of orthometric correction we get spheroidicorthometric correction. Clairauts formula gives normal gravity in relation to a spheroid, hence the name spheroidic-orthometric correction. One would think that Clairauts formula was used in the correction formula as a direct replacement for the weight, but it is not the case. It is likely to facilitate the calculations and to obtain a simple formula derived from correction on geoid. We can imagine that we look at change in weight by going from one point to another. The weight change as a result of the height, but also as a result of the banks. Because of the earth's shape will the distance between the potential surfaces shrink when we go north. Weight difference between two points in the same height can be found by derived Clairauts formula with respect to the width. We get: If we look at a stretch between two points, we can find the height of B by the height of the A and the leveling height differences in the B's vertical line along the potential surfaces. See Figure 2.3 Figure 2-3: System of leveling lines. We find the height of B by multiplying the H A with the relationship g 1 /g 2 which is gravity values on geoid in A 0 and B 0. The distance between two permanent signs of leveling is so short that we can put Page 6

7 geoid values for the board at the average of the gravity values in the endpoints, g M = (g 1 + g 2 ) / 2. We get: We set Δg = g 2 - g 1 and get: Mean height between the points (the last parentheses) may be called h and we get the correction part equal to: We have not measured the gravity values and Δg for a short distance is very small compared to g 2. Therefore we can replace g 2 with γ 0. Hence we set the gravity difference between two points and we sit back with the expression of spheroidic-orthometric correction: 3 NN1954 The intention of cohesion (equalization) of NN1954 was to create a network with orthometric heights, but because of the use of Clairauts formula for weight rather than measured weight values it has recently been thought that NN1954 is closer to normal levels than orthometric heights. NGO provides a description of the grid in geodetic work booklet 5, Precise leveling in Southern Norway. It provides a description of what the network includes and what was done by approximations. But also so much more that it is difficult to get an overview of what actually was done. In addition, it is incomplete in some areas. 3.1 Cohesion (equalization) in 1956 NN1954 is the term for both a height system and fundamental point that system refers itself to. Originally the fundamental point of elevation laid in Oslo, but before cohesion in 1956 it was moved to Tregde by Mandal. By the study of water level observations it was shown the fact that the land (ground) had raised in Oslo, and the point was moved to a place where the uplift was close to zero. Zero level was also depressed in relation to the old. The old zero point tied just to a water level gauge, while now we have several years of observations from seven gauges along whole the coast. Zero level was lowered 21.2 cm, which corresponded to mean water level average deviation from the old level. Page 7

8 In geodetic work, clip 5 states that the net in 1956 included the 3822 regular brands. There were measured 40 closed loops with a length of 7166 km and 27 open lines with a total length of 1302 km. The network was linked to seven tide gauges, Oslo, Nevlunghavn, Tregde, Stavanger, Bergen, Kjølsdal and Heimdal. There are only 38 loops on the map of the loops and their loop errors. Two loops are also mysteriously disappeared from the array of constraint equations that came with the clip 5. One of them is around Bergen and another around Bunnefjorden. Why these are omitted is not mentioned anywhere, but that was thus 38 and not 40 loops that were leveled. All maps I have seen of the net is cut just north of Trondheim. This I have seen as natural since the network includes southern Norway. Upon closer look at the polls, however, there are measurements up to Mo i Rana from 1926 and The opening lines of total length also implies that this line was included in cohesion (equalization). The network included therefore in 1956, southern Norway and Trøndelag and Nordland south of Mo i Rana. The network is based on measurements from All leveling values incurred a spheroidicorthometric correction before the loops were leveled under one free-like bonding for correlation method. After the loops were equalized, the opening lines hooked on. Water meters are equalized separately and in the end is the free network solution to fit any water level gauges. The weights are set equal to the inverse weight proportional to linear length. This does not respond to the international recommendations from IUGG from 1948, but it was so they had put the weights in most countries. The way was also easy and was found as good as the international recommendations. There was already known the Post-glacial Rebound, but nobody had any model for how this turned out. This was therefore not taken into account beyond the zero point was moved. 3.2 NN1954 today Today, the NN1954 includes several new lines and the old ones are transferred again. By re-leveling, there are many points that are not found and the new established. New measurements are not corrected for the Post-glacial Rebound, and even if you recently have had weight measurements, these have not been used either. Leveling values are used directly without any correction. The term NNN1957 (North-Norwegian null) lapsed in 1996 when it was found that the difference between the two systems was less than expected measurement inaccuracy between them. NNN1957 is therefore a part of today's NN1954. Page 8

9 4 Comparison In this study, there is used data from the Norwegian Mapping Authority's height database. It contains: EUREF89 coordinates and NN1954-height for all leveling permanent signs, including those through the years disappeared. EUREF89 coordinates can be up to 100 meter error. The history of permanent leveling signs, (when they are created, what it is based on, if they have disappeared, though they are unstable and possibly moved to a new height or new name) All original leveling measurements from 1916 until today Spheroidic-orthometric correction for all measurements A Post-glacial Rebound model according to J.S. Danielsen Generated gravity values for all permanent marks. One can extract geopotential difference or height differences between the two and two permanent marks from the base. It can be selected to correct for the Post-glacial Rebound and/or add spheroidic-orthometric correction. There are used three data sets from this base: A) Height differences with spheroidic-orthometric correction B) Height differences with uplifting and spheroidic-orthometric correction C) geopotential with Post-glacial Rebound correction. In addition, it also provided a feedback (answer file) with all points of the EUREF89 coordinates and NN1954-heights. Data sets and answer file did not contain the same number of points and some points had different names in different data sets. With almost 11,500 points, it was a great challenge to pick out common points. All leveling is done in Gemini Surveying 4.4. For each data set is the entire net cohesive (equalized) as a whole, with solid brand of water meter by Slow as the only known point. 4.1 Reconstruction of NN1954 In dataset A, the measurements after 1954 removed. The remaining measurements should correspond to the measurements of cohesion in NN1954 was reconstructed with a deviation <3 mm. A few points had larger deviations, but these had been points that later got a new height, or were found unstable. A few lines were measured twice by cohesion in These measurement series are included in several year intervals and uplifting has manifested itself. At the cohesion (equalization) of the 1956 it was only known that the land uplifted. Therefore the crucial point was exceeded. But we had no good model for the uplifting and hence could not correct it. In the doublemeasured strain there were probably the last polls seen as the most appropriate, and there were only those which were included in the cohesion (equalization). This is not mentioned in some places, but the loop errors at this cohesion (equalization) should match those from 1956, the oldest measurement in double-measured tensile omitted. Page 9

10 4.2 Ideal heights For accurate height determination in large areas there should be leveling out and then derived geopotential height. In a country like Norway where the uplifting is so strong, one should also correct for the Post-glacial Rebound. Post-glacial Rebound adjusted to geopotential is equalized and then are orthometric heights and normal heights derived. For an ideal calculation of orthometric heights, there was calculated mean gravity along the vertical line for each point from the resulting gravity values by formula 2-1. Gravity values should actually be measured, but it is not measured in all leveling permanent signs. Maps are based on measurements in some places that developed an interpolation program that can generate gravity in any desired position. These generated that the weights will correspond with the actual weights. Mean normal weight is calculated for each point from the point's position in Formula 2-2. Finally, the difference with an answer file intended. 4.3 The difference due to real orthometric heights In Figure 4-1 we see that the orthometric heights differ little from NN1954 in the lowlands, but the discrepancy becomes larger with increasing altitude. The deviations are even greater when we go north-east. On the coast of Trøndelag is the deviations as large as the Filefjell. Especially towards the Swedish border in Hedmark there are large deviations. 4.4 The difference to normal levels On the Figure 4-2 we see that the normal levels differ generally less from NN1954. The ratings reflect well in the southernmost part of South Norway, also in mountainous areas. But as the deviations have the same tendency as those for orthometric heights, it increases when we go north-east. Especially the area around Trysil and from Røros and towards the Swedish border is the large discrepancy. Although the deviations are consistently less than the previous figure, we cannot say that NN1954 is normal heights. Page 10

11 Figure 4-1: Difference between the orthometric heights and NN1954-heights Page 11

12 Figure 4-2: Difference between normal levels and NN1954-heights Page 12

13 4.5 Deformation of the grid NN1954 is not corrected for Post-glacial Rebound. The measurements were performed in a period of almost 40 years in the first round. Later several lines were added, but these have not been adjusted for the Post-glacial Rebound. NN1954 is therefore based on measurements over a period of more than 80 years. The great uplift we have in large parts of Norway has contributed greatly to the deformation of the network. We see that the deviations in Figure 4-1 and Figure 4-2 increases towards the Swedish border. If we look closely at J.S. Danielsen's model for Post-glacial Rebound in Figure 4-3, we see that the uplift also increases towards the Swedish border, and is greatest in the area around Trysil. Uplift is greatest where the deviations are greatest. The large deviations may be due to uplift. In Hedmark is the first poll conducted in the '30s. Uplift are 4-5 mm per year. At about 20 years (from 30's to 54) this went to about 9 cm. NN1954-heights is thus about 9 cm for low here compared to other parts of the network. In Hedmark, we have in the reality heights that relate to the 30 - century, while my ideal height refers to The largest deviations found from Røros and towards the Swedish border. There is also uplift 4-5 mm per year and the measurements are from At 34 it uplifted here over 15 cm! Figure 4-3: Post-glacial Rebound model for Scandinavia by J. S. Danielsen Page 13

14 Measurements that have come later have not been adjusted to the Post-glacial Rebound. Gravity Values have not been used even though they were available. Leveling values are used directly without any correction. The error that was made in respect of the Post-glacial Rebound at the cohesion of NN1954 will therefore propagate to the points that have come later. Some lines are a pure extension of the network. If these had been corrected for the Post-glacial rebound would not deviations become so large. The loose line of Hitra and Freya in Sør-Trøndelag (Southern Trøndelag) and from Foldereid to Vikna in Nordland (Northern Land) is an example of this. The lines are measured, respectively, in 1981 and With a Post-glacial Rebound of about 3.5 mm per year will amount to almost 10 cm, if we return to In Figure 4-1 we see that the deviations of the mentioned lines are in the range of cm. Adjusted to the Post-glacial Rebound would have been in the range 2-4 cm as ratings improve from the height of the area. Other lines are forced into the existing network. A Post-glacial Rebound correction back to 1954 on these measurements will not be sufficient since NN1954 network has no fixed reference period. Forcing the line into mean that the measurements are corrected to the deformed network, whatever the deformation caused. Line from the south end of Femunden and north Tufs Ingdalen to Os is an example of this. In the north, tied the line at a point measured in 1918, while the south is linked on a point first measured in Uplifting in the area will be a factor that deforms the net, but there are also other factors, such as unstable points. It seems that uplifting is the reason here why deviations are so large. 5 NN1954 corrected for Post-glacial Rebound By studies of the computational methods that underlie NN1954 it suggests orthometric elevations or normal levels, you have to take into account the uplift. The uplifting of the land disturbs the image so much that we cannot find any clear correlation otherwise. NN1954 is equalized again with all the measurements we have, but this time the measurements are corrected for Post-glacial Rebound, in addition to the spheroidic-orthometric correction. We then get a network of fixed reference period Page 14

15 Figure 5-1: Difference between ortometriske heights and corrected NN1954-heights Page 15

16 Figure 5-2: Difference between normal levels and corrected NN1954-heights Page 16

17 When we compare the orthometric heights and normal heights now, respectively Figure 5-1 and Figure 5-2, we see that the deviations are much smaller. The large deviations in Hedmark are gone. We see clearly that orthometric heights match well along with corrected NN1954-levels in the lowlands, but the deviation is systematically greater with increasing altitude. If we look at variation of height deviations with respect to height, it is even more evident. NN1954-corrected heights are too low in mountainous areas in relation to the real orthometric heights. We see a slight curve in Figure 5-3. The deviations are potentially dependent on the height. We adapt a different degree function and this fits very well. The standard deviation is 0.8 cm. R-sq, or R 2 is an expression of how well the feature explains the variation in the data. Here we see that the function explains 81.5% of the variation. R 2 is the same as the correlation of different potency. The correlation is here 0.9. Previously it has been shown that the difference between normal levels and ortometriske heights square depends on the height. This suggests that the corrected NN1954-levels are normal heights. Figure 5-3: Height deviation (orthometric heights - corrected NN1954-heights) variation with respect to adjusted NN1954-heights. Comparison of normal levels and corrected NN1954-heights in Figure 5-2 also indicates the same. We see that the deviations are within five centimeters of the entire area. The deviations are positive in the lowlands and are negative in the mountain areas. 6 Conclusion NN1954-heights are in mountainous areas generally too low in relation to the true orthometric heights. In the lowlands votes the better, but in areas where uplifting is great there are much less Page 17

18 NN1954-heights than the true orthometric heights. With the large deviations we have, we cannot say that NN1954-heights are orthometric heights. The deviations of normal levels are generally lower, particularly in mountainous areas. But the picture is complex, and in parts of the network are the deviations so large that we cannot say that NN1954-levels are normal heights either. The most deformed network is the closest to normal leveling calculations. A less deformed network with a reference period gives a more clear answer. Corrected NN1954- heights match well with orthometric heights in the lowlands, but is too young in the mountain areas. The match well with normal levels over the entire Web. We can therefore say that the computational methods that underlie NN1954 are almost a normal height calculation. The difference between normal levels and orthometric levels are quadratically dependent on the height. It turned out that the difference between the adjusted NN1954-heights and orthometric heights are also quadratic depending on the height. This is also a strong indication that the corrected NN1954-levels are about the same as normal heights. 7 Bibliography 1. Danielsen, J. S, 2001: A land uplift map of Fennoscandia Survey Review, 36, Heiskanen, W. A and Moritz, H, 2000: Physical Geodesy. W. H Freeman and Company. 3. International Association of Geodesy, 1950: Bulletin geodesique 1950, no 18, desember. 4. Mathisen, O. 1991: Den Gaussiske projeksjon 5. Mathisen, O. 1995: Forelesniger i geodetisk høydegrunnlag for landmålingsstudentene ved Norges landbrukshøgskole 6. Norges geografiske oppmåling, 1956: Presisjonsnivellement i Sør-Norge geodetiske arbeider, hefte Statens Kartverk 1997: Foreløpig utkast til rapport fra Høgdeutvalget 8. Statens Kartverk, 2001: Rapport nr 2 fra Høgdeutvalget 9. Statens Kartverk, 2002: Norges offisielle høydesystemer og referansenivåer Versjon Torge, W. 2001: Geodesy 3rd edition Page 18