1 The Future of Vertical Geodetic Control

G. Steinberg H. Papo Survey of Israel Technion, Israel Institute of

Abstract The phenomenal progress made by motorized leveling in the past two decades has not changed the status of vertical geodetic control (VGC) worldwide. Only a few countries have been able to establish and to maintain (re-observe at reasonably short time intervals) their national VGC networks at first order accuracies. The establishment of a national VGC network by precise leveling has been indeed a technical possibility for many years and for some countries it has been even economically affordable. We claim that today those national VGC networks are not indispensable. Due to their many drawbacks such as high cost, slow pace, obstacle-crossing impotence and monumentation problems, we believe that they are going to be replaced in the near future by a more attractive alternative. The alternative we envision is a national VGC network based on GPS measurements which forms an integral part of a 3-D spatial network. The reference surface is the WGS84 (or any future equivalent) ellipsoid and thus the system of heights is ellipsoidal and not orthometric. We believe that the switch from orthometric (OVGC) to ellipsoidal (EVGC) height systems at the national (or at the continental or even at the worldwide) level is imminent and inevitable. OVGC networks may continue to be used in the future but that only in the form of floating islands with a limited geographical extent. Viable candidates for such orthometric islands are regions and projects where the use of orthometric heights remains mandatory. There are cases where the island can be ellipsoidal in character or in other words form a densification layer beneath the national EVGC network. Typical examples of such orthometric or ellipsoidal islands are urban areas, low-shore areas, sensitive water system projects etc. Each island maintains its own independent datum based on those points through which it is attached to the national EVGC network. In Israel we are already experimenting with the above flexible scheme of vertical control. The national EGVC network is composed of two layers: a primary layer of ten permanent GPS stations at distances of 30-90 kilometers and a secondary layer of 150 extremely well monumented points (the G1 geodynamic network) at distances of 10-20 km. Four orthometric islands based on the above secondary layer are at various stages of construction: The Mediterranean island: a narrow strip of land along the coast covering an area of 2000 km2; The Mount Carmel island: with 67 points over an area of 600 km2; 2 The City of Eilat island: by the coast of The Red Sea with 20 points over an area of 20 km2; The City of Modyin island: and its environs (still in the early stages of planning).

1. Introduction Over the years has been, and still is, very proud of its precise leveling technique that can produce elevation differences between distant points on the earth surface with an unsurpassed accuracy of fractions of a millimeter. The problem with leveling has never been its accuracy. The main problem associated with leveling is the technique itself which is notoriously time consuming, too expensive in terms of manpower and extremely cumbersome in having to cover literally step by step the distance between the end points. Motorized leveling has solved the slow-rate problem but the price remains prohibitively high, and still lengthy meandering is needed in order to cross mountains, wide rives or lakes. On top of the above problems related to the leveling technique, vertical crustal motions literally “slap us in the face” by rendering obsolete the heights of many points only a short time after their measurement and adjustment (see [3] as an example for difficulties in interpreting results of precise leveling). Vis-a-vis leveling, GPS as a geodetic tool has been advancing in the past 10 or 15 years by leaps and bounds. In terms of accuracy its progress has been impressive although in the vertical direction it is still at the mercy of chronic poor-geometry, insufficiently well resolved tropospheric delays, multipath and antenna focus variations. For the time being GPS can not compete with precise leveling as far as accuracy is concerned. But in all other respects it is a sure winner: it is fast, cheap, can easily bypass almost any obstacle. It only remains to see GPS make a significant improvement of accuracy in the “up” direction too. There is one additional problem which concerns not so much us - geodesists and surveyors - but rather the so called “consumers of heights”. Even if the ellipsoidal (GPS borne) heights were ten times more accurate, there are going to be users who would prefer the "good, old..." and familiar orthometric heights. "Sensitivity to transition" analyses of applicability of the ellipsoidal heights in various scientific and engineering enterprises have been made and published (see [1]). The bottom line has being that although the transition is not going to be easy, in the majority of cases it would be possible and even advantageous to use ellipsoidal (geometric) heights instead of ordinary orthometric (physical) heights. In this paper we are dealing not so much with the above "non-geodetic" problem but rather with the establishment of a novel and unorthodox system of vertical geodetic control. Miracles that were apparently commonplace in biblical times don’t happen any more. The inevitable transition from orthometric to ellipsoidal vertical control systems is just one such miracle which requires lots of work and patience from all those involved - geodesists as well as 3 their "customers" the consumers of heights (see [7]). In the next section we present schematically our vision of that future national ellipsoidal vertical geodetic control that will be based on GPS measurements alone.

2. The ellipsoidal alternative As an alternative to the existing national orthometric vertical control (NOVC) networks we propose to declare the ellipsoid (the WGS84 ellipsoid or any suitable equivalent) as the only reference surface needed to define the position of a control point in all 3 dimensions, including heights (above the ellipsoidal surface). The realization of a comprehensive and accurate ellipsoidal vertical geodetic control (EVGC) network based on GPS measurements does constitute an integral part of a 3-D spatial control network of any given country. As we all know, the above is not a vision anymore. GPS measurements backed up by the IGS permanent stations array have reached a level that makes it possible for any country to establish a nationwide GPS-borne 3-D geodetic control network. The time required for and subsequent analysis is short and the cost is unbelievably low. As with the conventional horizontal and vertical control networks there is an advantage in constructing the EVGC net in a hierarchical order. Monumentation subtleties, optimal distances between points, types of GPS receivers and antennae, software sophistication, duration of measurement sessions - all have to be designed and coordinated in view of the well known principle of going “from the whole to the part”. A typical national EVGC network should consist of three to four control layers as follows: (a) The fundamental primary (datum) layer consists of points at distances of 100-250 km. Most or even all of those points should be permanent GPS stations. The datum of the fundamental layer should be based on fixed (nominal) ellipsoidal heights of those points. If kinematic or dynamic models of analysis are adopted, datum definition may become more complicated but still it is an easily manageable problem. Accuracy of the adjusted heights should be of the order of 6-8 mm. (b) The secondary layer consists of points 25-40 km apart, where relative accuracies in height are better than 1 cm. Location and monumentation of its points should be based on geometrical and geo-technical considerations. The first two layers are not intended for the occasional user but are there to provide control for lower level (densification) layers. (c) The third layer should consist of points at 6-10 km apart with relative accuracies still at the level of 1 cm. Due to the significantly shorter GPS vectors, a vertical accuracy of 1 cm can be obtained by relatively short GPS measurement sessions. Another characteristic that is 4 particularly important for the third layer of control is easy accessibility. An option that should be considered is to collocate between third layer control points and existing conventional vertical control points (particularly well monumented benchmarks). Whenever considered necessary, a lower (forth) layer of vertical control can be constructed to serve the specific needs for vertical control of a municipal area or that of a large industrial complex. Our proposal provides also a solution for height users who insist on having a forth layer of vertical control where heights are, however orthometric and not ellipsoidal. Such a mix- up is possible provided the boundaries of the "orthometric island" (OI) are clearly outlined and measurements and their processing are performed in a clear and consistent manner as proposed in the next section of this paper. Mixing physical (orthometric heights) and geometric (ellipsoidal heights) entities is not a new idea in Geodesy. In pre-1957 times (the first “sputnik” was put in orbit in 1957) national horizontal geodetic control networks were attached to the physical body of the earth by means of astronomical (physical) observations. In our vertical OI analogy each island will have its own independent datum - just as was the case in the past when each country had its own .

3. Orthometric islands Orthometric islands should be regarded as a temporary measure. They are intended to soften the shock of transition from the current orthometric heights system to the future ellipsoidal system. One can regard the OI as a compromise, a continuation of the present state of affairs with only one significant difference namely: each OI maintains an independent orthometric height datum. Thus there will be no need to maintain a consistent and up-to-date nationwide orthometric vertical control system. If a certain geographic area is marked for extensive crustal motions it may be declared an independent OI; it will have to be re-leveled frequently and be analyzed as a kinematic entity in order to determine time tagged heights. Orthometric islands may be the solution for cases where ellipsoidal heights are inapplicable. Typical examples of engineering enterprises where ellipsoidal heights are inconvenient or even impossible to use are large water system constructions, tunnel excavations with two-ends monitoring, construction in highly developed shore areas, open-ditch irrigation projects etc. Another reason for preferring or needing an OI may be due to local difficulties in collecting GPS measurements as for example in heavily wooded areas, in large cities, etc. In such cases only an OI can provide control for vertical surveys which are performed by levels, or total stations. No matter why we do establish a local OI system it should be always regarded as a secondary control which “draws” its datum from the national ellipsoidal control (NEC) GPS-borne network (see [6]). 5 Several variations and a number of problems can be envisioned and should be tackled as they develop: * There may be an overlap between two adjacent islands so that a number of points may belong to both islands. A hierarchical order may have to be set to take care of such dualities. * Certain points belonging to an island may have only orthometric heights i.e. (have not been visited by GPS). Their ellipsoidal heights will depend on the quality of the local geoid model. A typical orthometric island is composed of m points, which are distributed within a given region and are monumented so as to create a continuous local vertical control network. The OI is surveyed by conventional (classical or motorized) precise or second order leveling techniques. Part of the m OI control points (a subset of size k) belong also to the 3-D NEC network. Through those k points the OI is “attached” to the higher order NEC network. Those k attachment points are instrumental for defining the datum of the OI. In the remaining part of this section we describe the methodology of OI datum definition: (a) From the output of the NEC 3-D network adjustment (GPS measurements) we extract a

vector h0 (of size k) and a corresponding covariance matrix Qh0. (b) The best available geoid model (in the OI area) is employed to compute geoid undulations

for each one of the k points. The result is a vector u0 and an estimate of its covariance

matrix Qu0.

The two vectors are combined to create a vector of nominal orthometric heights (of size k): with a respective covariance matrix

QH0  Qh0  Qu0 . (c) The n (n>m) leveling measurements of the OI are adjusted by the method of parameters where the vector of m unknown orthometric heights H is partitioned into:

H0 - heights of the k “attachment” points.

H1 - heights of the remaining (m-k) points. The normal equations are formed as usual, keeping in mind the above partitioning:

N11 N10  H1 U1     =   . N01 N00 H0 U0

The H1 unknowns are eliminated from the normal equations system by substitution: -1 H1 = N11 U1 - N10 H0 . As a result we obtain the reduced system of normal equations: 6

N 00 H 0 = U0 where N = N - N N -1 N 00 00 01 11 10 . -1 U0 = U0 - N01 N11 U1

Datum of the orthometric heights within the OI is still to be determined. Due to lack of datum the rank of the above N00 k by k matrix is only (k-1). The rank deficiency of N00 [d = k-(k-1) = 1] means that there is a whole family of different H0 vectors all of which satisfy the reduced system of normal equations. Any two such vectors are related through Helmert's matrix E (k by 1):

H0j = H0i + E  ij where

ET = 1 1 1 ... 1

th th and ij is the vertical datum shift between the i and the j solutions. We define datum of the OI by the well known weighted free net (WFN) principle where out of the multitude of H0 solutions we select the one (and only) solution which minimizes the following quadratic form: T D PD D = min where

D = H0 - H0 and

-1 PD = QH0 .

It can be shown that the unique vector H0 , that satisfies the above minimum condition, satisfies also the following linear equations:

T T T T T E PDD  0 = E D = E H0 - E H0 = E H0 - H0 and finally

which are known as the weighted free net constraints of H0. In this case as the datum "defect" is of size one (d=1) the WFN constraints are in fact a single linear equation. The reduced normal equations bordered by the above WFN constraints form a system of full rank and result in a unique solution for H0: 7

1  T 1  H   N E U0  Q E E E U0   o    00      H0       T  1  E 0  H   T T   H       0  E E E 0   0  where  is the Lagrange multiplier of the WFN constraint.

The solution for H0 is thus:

1 H = Q U + E ETE H . 0 H 0 0 0

Matrix QH0 , the reflexive symmetrical generalized inverse of N00 ,that happens to be also the covariance matrix of H0 , is computed by the following expression as shown in [2]:

1 1 Q = N + EE T   EE T EE T E E T H0 00

4. Unorthodox vertical control in Israel Geodetic activities in Israel are led by the Survey of Israel (SOI). Active participants in those geodetic activities (mainly in research and development) are also the Technion in Haifa, the Tel Aviv University, the Geological Survey in Jerusalem and the Geophysical Institute in Holon. Ideas presented in this paper, that were published earlier in [5], are not yet fully accepted in Israel. Understandably, they have not been adopted as a basis for the official geodetic policy of the state. What we intend to do in this section is to describe a few existing components of our still nonexistent ellipsoidal vertical geodetic control system. Pending an official adoption of our proposals those components could be easily put together with little additional work so as to form a living example of our "future" vertical geodetic control system. The vertical geodetic control network in Israel is conventional i.e. it is based on precise leveling. The primary net has been re-leveled and readjusted recently in order to update the rapidly changing heights of its points. Due to budget cuts it seems that the SOI will not be able to embark in the foreseeable future in additional massive re-leveling of the primary net. The inevitable effect of the above situation is that in just a few years the adjusted heights will become obsolete and the nationwide vertical control network will become a fiction. A possible remedy at the moment seems to be the marking of a number of orthometric islands in regions and municipal areas where budgets for re-leveling could be obtained. Our NEC exists in 3-D in the form of a dense network of some 150 extremely well monumented control points - the so-called G1 geodynamic network. The GPS measurements campaign of the G1 lasted a period of little more than half a year (1996-97) where each point was occupied by a dual frequency P-code GPS receiver for at least two 24-hours sessions. The accuracy of the G1 network is estimated at the level (one sigma) of 4-5 mm in horizontal position and about 10 mm 8 in ellipsoidal heights. Three (out of the seven presently operating) permanent GPS stations were active during all of the G1 measurement campaign and provided primary geodetic control for the adjustment of the G1 net. By its accuracy and also by the distribution of its points (see figure 1) the G1 can very effectively serve as a NEC network for the State of Israel. In the remaining part of this section we describe three existing orthometric islands (OI) which are in the process of being connected to the NEC by GPS measurements. A forth island (still in the design stages) will serve as a laboratory for experimenting with "pure" ellipsoidal height control. (a) The Mediterranean-Sea island Part of the existing primary conventional (orthometric) vertical control network along the Mediterranean Sea shores of the State of Israel forms a narrow strip roughly 10 km wide. (see fig. 2). The Med-Sea island will be re-leveled, readjusted and attached to the NEC. In fact, due to the recent (1997-98) readjustment mentioned above, all that has to be done at the moment is to attach a number of its points (benchmarks) to the NEC. The geoid model which is an indispensable part of the attachment process (see section 3) is still of insufficient quality and a special effort will have to be made for its eventual improvement before it could be used to attach the island to the NEC. (b) The Mount Carmel Island As part of a long range research project for improving the geoid model of the State of Israel (see [4]), an area of some 600 km2 has been surveyed intensively by GPS and by short-leg precise trigonometric leveling. The 67 control points form a contiguous island, which overlaps with the Med-Sea island. In this case as the geoid model is of a superior quality with an accuracy of 4 cm (one sigma) all that remains to be done is to attach the Mount Carmel island to the G1 network (NEC) by GPS measurements. (c) The Eilat (Red Sea) Island This tiny (20 Km2) island has been set as a pilot project to explore possibilities for the creation of small “municipal” OI. A dense network of control points set at ½ km distance has been surveyed by second order leveling as well as by GPS. The island is already attached to the NEC (G1 net) and its local geoid model, due to the extremely high density of surveyed points, is accurate to better than 1 cm (one sigma). (d) The city and environs of Modyin island This island is still on the “drawing table”. The new city of Modyin and its environs are located far from the Mediterranean Sea. As such they seem to be ideal for conducting an experiment with “ellipsoidal” islands (EI). In such a case, the island is nothing else but a densification of vertical ellipsoidal control within the islands boundaries. The local geoid 9 model will have to be improved dramatically on the basis of our experience with (b) and (c). So far from our limited experience with the compilation of unorthodox vertical control in Israel we have learned that it can be extremely flexible and adaptive.

5. Summary In trying to sum up the ideas that were presented in this paper we would like to make a number of observations taking the position of a “devil’s advocate” of our proposal. We list items which are instrumental for the realization of an ellipsoidal vertical control system and which, at least at the moment, seem to be missing, inadequate or prohibitively expensive and hard to get: (a) GPS is still unable to provide reliable and accurate to 3-4 mm ellipsoidal height differences. Only by using chock-ring antennae and after long measurement sessions (12 hours and up) can we hope to get closer to that accuracy goal. (b) High accuracy (1 cm sigma) local geoid models (within the range of an orthometric island) are still a hard to get commodity. Development of models at that level of accuracy requires extensive fieldwork in addition to the integration of gravity and topographic data. Airborne gravimetry may hold a solution of the precision geoid problem by providing the necessary data in remote or inaccessible areas. (c) The community of height users is still reluctant to part with the orthometric heights to which it has been accustomed so far. The proper way of "making friends" is to let them have a fair chance to work with ellipsoidal heights and learn from their own experience. We, the ellipsoidal-height providers, will have to follow them up and help in solving the many smaller problems on the way, as they arise. In spite of the above listed difficulties we think that the transition from orthometric to ellipsoidal vertical control is inevitable and in fact it is already happening. The final format may be somewhat different from the model that was described in sections 2 and 3 above. We believe that it is our duty to explore honestly the new ellipsoidal alternative for vertical control. To do that, first of all, we have to suppress our inherent professional prejudices and suspicions. We, the small community of thinking-ahead-of-our-times individuals, have to prepare for the inevitable transition from orthometric to ellipsoidal vertical control by intensifying our research and by gaining experience with the new media in small scale experiments as well as in more ambitious pilot projects. Extensive research during a quarter of a century has brought up excellent results in improving conventional leveling. Let us hope that a similar wave of enthusiasm coupled with plenty of talent will be invested in studying the new technique and will bring us soon to the vertical geodetic control of the future. 10 References [1] Kaula W.M., (1987), "The Need for Vertical Control", Surveying and Mapping, Vol. 47, No. 1, pp. 57-64. [2] Koch K.R., (1988), "Parameter Estimation and Hypothesis Testing in Linear Models", Springer Verlag, Berlin, Heidelberg. [3] Makinen J. and Saaranen V., (1998), "Determination of Post-Glacial Land Uplift from the Three Precise Levellings in Finland", Journal of Geodesy, 72, pp. 516-529. [4] Sharni D., Papo H. and Forrai J., (1998), “The Geoid in Israel: Haifa Pilot”, in "Second Continental Workshop on the Geoid in Europe", Budapest, Hungary. [5] Steinberg G. and Papo H., (1998), “Ellipsoidal Heights: The Future of Vertical Geodetic Control”, GPS World, Vol.9 , No.2, pp. 41-43. [6] Wells D., Kleusberg A. and Vanicek P., (1996), "A Seamless Vertical Reference Surface for Acquisition Management, and Display of Hydrographic Data", Report submitted to The Canadian Hydrographic Service. [7] Zilkoski D.B., (1993), "A Strategy for an Orderly Transition from Leveling-Derived Orthometric Heights to GPS-Derived Orthometric Heights", ACSM/ASPRS Annual Convention, New Orleans, Louisiana, USA.