The book can be used as a general textbook for undergraduates studying geomatics and survejing and mapping in higher education institutions. For technicians who are engaged in geomatic and surveying engineering, the book is strongly recommended as a basic and useful reference guide.
Zhiping Lu, ssscenter Shubo Qiao, qsb shao.
Du kanske gillar. Spara som favorit. Skickas inom vardagar. Several more realistic formulations of the problem have been introduced into the geodetic literature. After reviewing them, an ultimate formulation is attempted where only local data are given stemming from levelling, gravimetry, classical geodetic network observations, combined with global GPS-like observations.
For the sake of simplicity we also assume a model of a rigid earth uniformly rotating around an axis fixed in space as well as with respect to its body and we stipulate that all the observations are reduced for the solid tidal effects, apart from the constant the so-called Honkasalo term. G can then be univocally identified either by specifying an a priori value W 0 , such that the stated conditions are satisfied, or by claiming that the equipotential surface at hand is the one that passes through a point P physically placed close to the sea surface e.
Once G has been somehow defined we must have a suitable mathematical representation for it and a sound mathematical theory capable of retrieving this surface, which we use e. The description of G is done by the so-called geoid undulation N and we have to stress here that this is a function of both the point P on the geoid and the reference ellipsoid E with respect to which we compute it; N is in fact the height of P on E reckoned along the line orthogonal to E , passing through P.
It seems more or less one of the ordinary miracles of geodesy that the problem of determining the height datum and the ellipsoid E from surface data of any kind can be split into two parts when we reason in the so-called spherical approximation. Nevertheless this approach is not anymore sufficient when we come to the centimetre accuracy which is nowadays possible for both the point positioning in space by GPS and other space techniques and gravimetric geoid computations.
In this paper we try to clarify how these two facts are intermingled at an ellipsoidal approximation level, also taking into account that we have to face and solve this problem every time we compute a high resolution geoid and not only for global models, as it was devised in earlier formulations. We shall conduct our derivations by ordinary approximation techniques trying to keep an accuracy of at least 1 cm in positions and at least 0. This defines what we mean by approximation throughout the paper.
To simplify the reading of the paper all symbols used are listed in Table 1. It seems to the authors that the evolution of the definition of the height datum problem can be characterized by the following steps:.
Note that cf. While P is sweeping the earth's surface, Q describes the so-called telluroid. At that time the possibility of a true global computation of the geoid was still in the realm of theory, yet it is interesting to observe that in principle no additive information was needed to solve the problem, which might appear curious since we have suppressed one information, namely W 0 , from the given data.
All the space geodetic observations can be analysed together, as the International GPS Service does for the international GPS network, providing geometric positions of the stations in a unified geodetic datum. Therefore, in particular, at these points we know the ellipsoidal heights in a consistent geocentric geodetic datum.
At the same points we know as well the approximate heights h Q is , derived from. Additionally, when considering a unified height datum over large areas e. North America or Europe one should be aware that probably significant distortions have been introduced to connect partial levelling networks. Since we shall return to this item at the end of the paragraph, we won't dwell on it here. The approach in d , presented in Lehman , is the only one, for the moment, taking into account that typically geodetic data on land and ocean differ significantly leading, for the globe, to the formulation of mixed BVP's, the so-called Altimetry Gravimetry problems.
The analysis is more directed to study the uniqueness of the solution of the modified BVP's so we shall not go into details here apart from underlining that in the future, considering the global problem in the form of mixed BVP is mandatory if we want to be realistic. The local approach to the height datum determination, e , has been proposed by Milbert and Forsberg This approach is undoubtedly appealing because it is reducing the problem to the use of a realistic data set and also because it exploits the full power of modern approaches to the geoid estimation, like the so-called Least Squares Collocation Theory cf.
This approach has to be pursued even if data are given in a single area, if, afterwards, we want to be able to compute the transformation between the local geodetic datum and a geocentric datum. Otherwise our height anomalies will all be known up to an arbitrary almost constant bias. We come now to a precise definition of the problem we want to treat in this paper Item f.
For the sake of simplicity we shall treat it with only two geodetic datums, one geocentric, based on the ellipsoid E 0 , one local, based on an ellipsoid E L rototranslated with respect to E 0. Hereafter we shall use the indexes 0 and L for quantities referring to E 0 and E L respectively.
Only 5 left! Since reference datums can have different radii and different center points, a specific point on the Earth can have substantially different coordinates depending on the datum used to make the measurement. Gravitational space, surfaces, elevation. Recalling eq. There are a number of ways of initializing an OGRSpatialReference object to a valid coordinate reference system. Positioning of points.
To be more specific in the rest of the paper we shall adopt the approximate formulae in Table 2 , which are well suited for our purposes, especially to compute orders of magnitude. It follows that if we want to write and use the BVP theory for an anomalous potential, it is safer to do it for T 0 rather than for T L , though the final approximations implied are known to be very small.
So strictly speaking eq. Nevertheless the knowledge of a global model T M and of a residual height model of the terrain, with the corresponding residual potential correction T RTC , can help in finding a very acceptable solution by a so-called remove-restore concept cf. Moritz At this point one can apply any solution method, from the use of Stokes' function, to the use of a stochastic approximation technique like collocation cf. As we said in Section 2, geodetic space observations, in particular in GPS observations which are the most frequent, are able to provide us with the geometric positions of points S with respect to a geometric ellipsoid E 0.
The position of P is collected in the vector r 0 which we consider as the outcome of the space observation. If this is not the case, i. As it is obvious 51 is certainly suited to determine. In Section 3 we have established linearized and suitably approximated according to our definition at the end of Section 1 observation equations, involving the unknowns of our problem, namely:.
T 0 P : anomalous potential with respect to a normal field attached to a geocentric reference ellipsoid,. The solution should be achieved on a rigourous basis, as observed at the end of point b in Section 2, by including for all the equation a noise model and treating them as in a form of an overdetermined boundary value problem.
An equivalent solution can be found by eliminating part of the unknowns and part of the equations. This can be done by using 53 , to eliminate T 0 , i. Naturally the main road stays for us in using the approach presented here. Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide. Sign In or Create an Account. Sign In. Advanced Search. Article Navigation. Close mobile search navigation Article Navigation. Volume Article Contents.
http://dev.vankaarstotservet.nl/9-azithromycin-y.php Oxford Academic. Google Scholar. Cite Citation. Permissions Icon Permissions. Summary The height datum problem is present in the geodetic literature since the times of Pizzetti, when it was realized that as only differences of the gravity potential can be derived from measurements, there was still one global parameter to be settled in order to determine a global model. From the above quantities one can compute the constant cf.
At this point if one adds the spherical coefficient J 2 , again fairly well known from its large effects on satellite dynamics, one has access to the eccentricity of the ellipsoid, , which can be computed by solving the exact equation. Open in new tab Download slide. To conclude this section, let us claim that since following Molodensky's line of thought one can always write cf.