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1.
Charles F. F. Karney 《Journal of Geodesy》2011,85(8):475-485
Implementations of two algorithms for the transverse Mercator projection are described; these achieve accuracies close to
machine precision. One is based on the exact equations of Thompson and Lee and the other uses an extension of Krüger’s series
for the mapping to higher order. The exact method provides an accuracy of 9 nm over the entire ellipsoid, while the errors
in the series method are less than 5 nm within 3900 km of the central meridian. In each case, the meridian convergence and
scale are also computed with similar accuracy. The speed of the series method is competitive with other less accurate algorithms
and the exact method is about five times slower. 相似文献
2.
G. Blaha 《Journal of Geodesy》1978,52(1):19-23
A simple formula is presented giving the value of γ−γ
r
to better than 0.001 mgal associated with an arbitrary reference ellipsoid, where γ is the normal gravity and γ
r
is its radial component. Further simplifications of this formula are possible, depending on the desired accuracy. Since in
the actual field g−gr equals γ−γ
r
to a good approximation, this formula makes it possible to work in terms of gr rather than in terms of the measured quantity g. Such a choice is attractive mainly because the spherical harmonic expansion
of gr is very simple. 相似文献
3.
《测量评论》2013,45(87):12-17
AbstractThe excuse for yet another paper on the Transverse Mercator projection, which has already received what should be more than its fair share of space in this Review, can only be that there is a fresh viewpoint to offer. It is the purpose of this paper to show that there are, in fact, two “Transverse Mercator” projections of the spheroid, of which one has hitherto almost escaped notice. 相似文献
4.
As a conformal mapping of the sphere S 2
R
or of the ellipsoid of revolution E 2
A
,
B
the Mercator projection maps the equator equidistantly while the transverse Mercator projection maps the transverse metaequator,
the meridian of reference, with equidistance. Accordingly, the Mercator projection is very well suited to geographic regions
which extend east-west along the equator; in contrast, the transverse Mercator projection is appropriate for those regions
which have a south-north extension. Like the optimal transverse Mercator projection known as the Universal Transverse Mercator Projection (UTM), which maps the meridian of reference Λ0 with an optimal dilatation factor &ρcirc;=0.999 578 with respect to the World Geodetic Reference System WGS 84 and a strip [Λ0−Λ
W
,Λ0 + Λ
E
]×[Φ
S
,Φ
N
]= [−3.5∘,+3.5∘]×[−80∘,+84∘], we construct an optimal dilatation factor ρ for the optimal Mercator projection, summarized as the Universal Mercator Projection (UM), and an optimal dilatation factor ρ0 for the optimal polycylindric projection for various strip widths which maps parallel circles Φ0 equidistantly except for a dilatation factor ρ0, summarized as the Universal Polycylindric Projection (UPC). It turns out that the optimal dilatation factors are independent of the longitudinal extension of the strip and depend
only on the latitude Φ0 of the parallel circle of reference and the southern and northern extension, namely the latitudes Φ
S
and Φ
N
, of the strip. For instance, for a strip [Φ
S
,Φ
N
]= [−1.5∘,+1.5∘] along the equator Φ0=0, the optimal Mercator projection with respect to WGS 84 is characterized by an optimal dilatation factor &ρcirc;=0.999 887 (strip width 3∘). For other strip widths and different choices of the parallel circle of reference Φ0, precise optimal dilatation factors are given. Finally the UPC for the geographic region of Indonesia is presented as an
example.
Received: 17 December 1997 / Accepted: 15 August 1997 相似文献
5.
While the standardMercator projection / transverse Mercator projecton maps the equator / the transverse metaequator equivalent to the meridian of referenceequidistantly, theoblique Mercator projection aims at aconformal mapping of the ellipsoid of revolution constraint to anequidistant mapping of an oblique metaequator. Obliqueness is determined by the extension of the area to be mapped, e.g. determined by the inclination of satellite orbits: Satellite cameras map the area just under the orbit geometry. Here we derive themapping equations of theoblique Mercator projection being characterized to beconformal andequidistant on the oblique metaequator extending results ofM. Hotine (1946, 1947). 相似文献
6.
《测量评论》2013,45(61):267-271
AbstractSome publications that have dealt with the question of convergence of meridians seem, to the present writer, to be clouded with misconception, and these notes are intended to clarify some points of apparent obscurity. For instance, A. E. Young, in “Some Investigations in the Theory of Map Projections”, I920, devoted a short chapter to the subject, and appeared surprised to find that the convergence on the Transverse Mercator projection differs from the spheroidal convergence; the explanation which he advanced can be shown to be faulty. Captain G. T. McCaw, in E.S.R., v, 35, 285, derived an expression for the Transverse Mercator convergence which is equal to the spheroidal convergence, and described this as “a result which might be expected in an orthomorphic system”. Perhaps McCaw did not intend his remark to be so interpreted, but it seems to imply that the convergence on any orthomorphic projection should be equal to the spheroidal convergence, and it is easily demonstrated that this is not so. Also, in the second edition of “Survey Computations” there is given a formula for the convergence on the Cassini projection which is identical, as far as it goes, with that given for the Transverse Mercator, while the Cassini convergence as given by Young is actually the spheroidal convergence. Obviously, there is some confusion somewhere, and it is small wonder that Young prefaced his remarks with the admission that the subject had always presented some difficulty to him. 相似文献
7.
《测量评论》2013,45(58):142-152
AbstractIn January 1940, in a paper entitled “The Transverse Mercator Projection: A Critical Examination” (E.S.R., v, 35, 285), the late Captain G. T. McCaw obtained expressions for the co-ordinates of a point on the Transverse Mercator projection of the spheroid which appeared to cast suspicion on the results originally derived by Gauss. McCaw considered, in fact, that his expressions gave the true measures of the co-ordinates, and that the Gauss method contained some invalidity. He requested readers to report any flaw that might be discovered in his work, but apparently no such flaw had been detected at the time of his death. It can be shown, however, that the invalidities are in McCaw's methods, and there seems no reason for doubting the results derived by the Gauss method. 相似文献
8.
《测量评论》2013,45(60):221-227
AbstractIn a previous article in this Review, the writer endeavoured to show that chains of minor triangulation could be adjusted by plane rectangular co-ordinates ignoring the spherical form of the earth with little loss of accuracy, provided that the two ends were held fixed in position. It was demonstrated that the plane co-ordinates produced by the rigorous adjustment between the fixed starting and closing sides, differ by only a comparatively small amount from the projection co-ordinates produced by a rigorous adjustment on the Transverse Mercator projection. The saving in time when computing by plane co-ordinates as opposed to rigorous computation on the projection by any method will be apparent to any computer with experience of both methods. 相似文献
9.
AbstractI. Introduction.—Map projection is a branch of applied mathematics which owes much to J. H. Lambert (v. this Review, i, 2, 91). In his “Beyträge zum Gebrauche der Mathematik und deren Anwendung” (Berlin, 1772) he arrived at a form of projection whereof the Transverse Mercator is a special case, and pointed out that this special case is adapted to a country of great extent in latitude but of small longitudinal width. Germain (“Traité des Projections”, Paris, 1865) described it as the Projection cylindrique orthomorphe de Lambert, but he also introduced the name Projection de Mercator transverse or renversée; he shows that Lambert's treatment of the projection was remarkably simple. 相似文献
10.
AbstractWhen computing and adjusting traverses or secondary and tertiary triangulation in countries to which the Transverse Mercator projection has been applied, it is often more convenient to work directly in terms of rectangular co-ordinates on the projection system than it is to work in terms of geographical coordinates and then convert these later on into rectangulars. The Transverse Mercator projection is designed in the first place to cover a country whose principal extent is in latitude and hence work on it is generally confined to a belt, or helts, in which the extent of longitude on either side of the central meridian is so limited as seldom to exceed a width of much more than about 200 miles. 相似文献
11.
AbstractIn January 1938 the writer decided against holding up for more years some work on the Transverse Mercator Projection (E.S.R., 27, 275). The extension to the spheroid was not then complete, nor is the present paper to be regarded as a logical continuance. It is first proposed to show the results of “transplanting” orthomorphically upon the spheroid a spherical configuration forming a graticule. 相似文献
12.
The Somigliana–Pizzetti gravity field (the International gravity formula), namely the gravity field of the level ellipsoid
(the International Reference Ellipsoid), is derived to the sub-nanoGal accuracy level in order to fulfil the demands of modern
gravimetry (absolute gravimeters, super conducting gravimeters, atomic gravimeters). Equations (53), (54) and (59) summarise
Somigliana–Pizzetti gravity Γ(φ,u) as a function of Jacobi spheroidal latitude φ and height u to the order ?(10−10 Gal), and Γ(B,H) as a function of Gauss (surface normal) ellipsoidal latitude B and height H to the order ?(10−10 Gal) as determined by GPS (`global problem solver'). Within the test area of the state of Baden-Württemberg, Somigliana–Pizzetti
gravity disturbances of an average of 25.452 mGal were produced. Computer programs for an operational application of the new
international gravity formula with (L,B,H) or (λ,φ,u) coordinate inputs to a sub-nanoGal level of accuracy are available on the Internet.
Received: 23 June 2000 / Accepted: 2 January 2001 相似文献
13.
The spacetime gravitational field of a deformable body 总被引：3，自引：0，他引：3
The high-resolution analysis of orbit perturbations of terrestrial artificial satellites has documented that the eigengravitation
of a massive body like the Earth changes in time, namely with periodic and aperiodic constituents. For the space-time variation
of the gravitational field the action of internal and external volume as well as surface forces on a deformable massive body
are responsible. Free of any assumption on the symmetry of the constitution of the deformable body we review the incremental
spatial (“Eulerian”) and material (“Lagrangean”) gravitational field equations, in particular the source terms (two constituents:
the divergence of the displacement field as well as the projection of the displacement field onto the gradient of the reference
mass density function) and the `jump conditions' at the boundary surface of the body as well as at internal interfaces both
in linear approximation. A spherical harmonic expansion in terms of multipoles of the incremental Eulerian gravitational potential
is presented. Three types of spherical multipoles are identified, namely the dilatation multipoles, the transport displacement
multipoles and those multipoles which are generated by mass condensation onto the boundary reference surface or internal interfaces.
The degree-one term has been identified as non-zero, thus as a “dipole moment” being responsible for the varying position
of the deformable body's mass centre. Finally, for those deformable bodies which enjoy a spherically symmetric constitution,
emphasis is on the functional relation between Green functions, namely between Fourier-/ Laplace-transformed volume versus
surface Love-Shida functions (h(r),l(r) versus h
′(r),l
′(r)) and Love functions k(r) versus k
′(r). The functional relation is numerically tested for an active tidal force/potential and an active loading force/potential,
proving an excellent agreement with experimental results.
Received: December 1995 / Accepted: 1 February 1997 相似文献
14.
Low-degree earth deformation from reprocessed GPS observations 总被引：3，自引：1，他引：2
Mathias Fritsche R. Dietrich A. Rülke M. Rothacher P. Steigenberger 《GPS Solutions》2010,14(2):165-175
Surface mass variations of low spherical harmonic degree are derived from residual displacements of continuously tracking
global positioning system (GPS) sites. Reprocessed GPS observations of 14 years are adjusted to obtain surface load coefficients
up to degree n
max = 6 together with station positions and velocities from a rigorous parameter combination. Amplitude and phase estimates of
the degree-1 annual variations are partly in good agreement with previously published results, but also show interannual differences
of up to 2 mm and about 30 days, respectively. The results of this paper reveal significant impacts from different GPS observation
modeling approaches on estimated degree-1 coefficients. We obtain displacements of the center of figure (CF) relative to the
center of mass (CM), Δr
CF–CM, that differ by about 10 mm in maximum when compared to those of the commonly used coordinate residual approach. Neglected
higher-order ionospheric terms are found to induce artificial seasonal and long-term variations especially for the z-component of Δr
CF–CM. Daily degree-1 estimates are examined in the frequency domain to assess alias contributions from model deficiencies with
regard to satellite orbits. Finally, we directly compare our estimated low-degree surface load coefficients with recent results
that involve data from the Gravity Recovery and Climate Experiment (GRACE) satellite mission. 相似文献
15.
《测量评论》2013,45(26):230-234
AbstractTHE resolutions and pious hopes (væux) passed by the International Union of Geodesy and Geophysics at Edinburgh in September 1936 have just been circulated in a formidable document of 8 pages and XXIX commandments. Of these, two affect the Cape-to-Cairo line particularly and they seem to deserve special study. The first of these, Number III—on systems of Projections—applies the meridional strips of the Transverse Mercator Projection apparently to all maps, topographical as well as cadastral. 相似文献
16.
Elliptical harmonic series and the original Stokes problem with the boundary of the reference ellipsoid 总被引：5，自引：0，他引：5
Since the earth is closer to a revolving ellipsoid than a sphere, it is very important to study directly the original model of the Stokes' BVP
on the reference ellipsoid, where denotes the reference ellipsoid, is the Somigliana normal gravity, andh is the outer normal direction of. This paper deals with: 1) simplification of the above BVP under preserving accuracy to
, 2) derivation of computational formula of the elliptical harmonic series, 3) solving the BVP by the elliptical harmonic series, and 4) providing a principle for finding the elliptical harmonic model of the earth's gravity field from the spherical harmonic coefficients ofg. All results given in the paper have the same accuracy as the original BVP, that is, the accuracy of the BVP is theoretically preserved in each derivation step. 相似文献
17.
《测量评论》2013,45(32):85-89
AbstractThe necessity of transforming rectangular co-ordinates from one system of projection to another may arise from, various causes, One case, for example, with which the present writer is concerned involves the transformation, to the standard belt now in use, of the co-ordinates of some hundreds of points of a long existing triangulation projected a quarter of a, century ago on a, belt of Transverse Mercator projection, In this case conversion is complicated by the fact that the spheroid used in the original computation differs from that now adopted, and, also, the geodetic datums are not the same, The case in fact approaches the most general that can occur in practice, One step in one solution of this problem, however, is of perhaps wider Interest: that is, the transformation from one belt of Transverse Mercator projection to another when the spheroids and datums are identical. It is this special case which will be discussed here. 相似文献
18.
The differential equations which generate a general conformal mapping of a two-dimensional Riemann manifold found by Korn
and Lichtenstein are reviewed. The Korn–Lichtenstein equations subject to the integrability conditions of type vectorial Laplace–Beltrami
equations are solved for the geometry of an ellipsoid of revolution (International Reference Ellipsoid), specifically in the
function space of bivariate polynomials in terms of surface normal ellipsoidal longitude and ellipsoidal latitude. The related
coefficient constraints are collected in two corollaries. We present the constraints to the general solution of the Korn–Lichtenstein
equations which directly generates Gau?–Krüger conformal coordinates as well as the Universal Transverse Mercator Projection
(UTM) avoiding any intermediate isometric coordinate representation. Namely, the equidistant mapping of a meridian of reference
generates the constraints in question. Finally, the detailed computation of the solution is given in terms of bivariate polynomials
up to degree five with coefficients listed in closed form.
Received: 3 June 1997 / Accepted: 17 November 1997 相似文献
19.
《测量评论》2013,45(21):417-422
AbstractThe Transverse Mercator Projection is also called the Conformal of Gauss since it was devised by him in the early part of the nineteenth century in connexion with the Triangulation of Hanover. It belongs to the class of cylindrical orthomorphic projections. That is to say, the Earth's surface, or part thereof, is developed on the surface of a cylinder, and there is practically no angular distortion, an angle on the surface of the Earth being represented on the map by almost precisely the same angle. The representation of meridians and parallels, for instance, shows them intersecting at right angles as they actually do on the Earth's surface; but this orthotomic condition, though essential, is not in itself sufficient for orthomorphism. 相似文献
20.
In an attempt to model regular variations of the ionosphere, the least-squares harmonic estimation is applied to the time
series of the total electron contents (TEC) provided by the JPL analysis center. Multivariate and modulated harmonic estimation
spectra are introduced and estimated for the series to detect the regular and modulated dominant frequencies of the periodic
patterns. Two significant periodic patterns are the diurnal and annual signals with periods of 24/n hours and 365.25/n days (n = 1, 2, …), which are the Fourier series decomposition of the regular daily and yearly periodic variations of the ionosphere.
The spectrum shows a cluster of periods near 27 days, thereby indicating irregularities at this solar cycle period. A series
of peaks, with periods close to the diurnal signal and its harmonics, are evident in the spectrum. In fact, the daily signal
harmonics of ω
i
= 2πi are modulated with the annual signal harmonics of ω
j
= 2πj/365.25 as ω
ijM
= 2πi(1 ± j/365.25i). Among them, at low and midlatitudes, the largest variations belong to the diurnal signal modulated to the semiannual signal.
Some preliminary results on the modulated part are presented. The maximum ranges of the modulated daily signal are ±15 TECU
and ±6 TECU at high and low solar periods, respectively. A model consisting of purely harmonic functions plus modulated ones
is capable of studying known regular anomalies of the ionosphere, which is currently in progress. 相似文献