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1.
Urho A. Rauhala 《Journal of Geodesy》1979,53(4):317-342
Array algebra forms the general base of fast transforms and multilinear algebra making rigorous solutions of a large number
(millions) of parameters computationally feasible. Loop inverses are operators solving the problem of general matrix inverses.
Their derivation starts from the inconsistent linear equations
by a parameter exchangeX→L
0, where
X is a set of unknown observables,A
0 forming a basis of the so called “problem space”. The resulting full rank design matrix of parameters L0 and its ℓ-inverse reveal properties speeding the computational least squares solution
expressed in observed values
. The loop inverses are found by the back substitution expressing ∧X in terms ofL through
. Ifp=rank (A) ≤n, this chain operator creates the pseudoinverseA
+. The idea of loop inverses and array algebra started in the late60's from the further specialized case,p=n=rank (A), where the loop inverse A
0
−1
(AA
0
−1
)ℓ reduces into the ℓ-inverse Aℓ=(ATA)−1AT. The physical interpretation of the design matrixA A
0
−1
as an interpolator, associated with the parametersL
0, and the consideration of its multidimensional version has resulted in extended rules of matrix and tensor calculus and mathematical
statistics called array algebra. 相似文献
2.
O. Remmer 《Journal of Geodesy》1971,45(1):29-36
As it has been shown by Kubik it is possible to get an estimate,
, of the reciprocal of the weight-matrix in an adjustment problem. If we want to see whether this new estimate
differssignificantly from our a priori valueQ
0 it is necessary to know the distribution function of the elements
, the
’s being the elements of
. This distribution is found in the present article and it is shown that it is not identical with any of the distributions
well known from statistical textbooks. Furthermore a way of computing this new distribution is presented. Finally the connection
with the chi-square distribution is explored and it is proved that the chi-square-distribution may be used as an approximation
for a large number of over-determinations. 相似文献
3.
The resolution of a nonlinear parametric adjustment model is addressed through an isomorphic geometrical setup with tensor
structure and notation, represented by a u-dimensional “model surface” embedded in a flat n-dimensional “observational space”.
Then observations correspond to the observational-space coordinates of the pointQ, theu initial parameters correspond to the model-surface coordinates of the “initial” pointP, and theu adjusted parameters correspond to the model-surface coordinates of the “least-squares” point
. The least-squares criterion results in a minimum-distance property implying that the vector
Q must be orthogonal to the model surface. The geometrical setup leads to the solution of modified normal equations, characterized
by a positive-definite matrix. The latter contains second-order and, optionally, thirdorder partial derivatives of the observables
with respect to the parameters. This approach significantly shortens the convergence process as compared to the standard (linearized)
method. 相似文献
4.
Lars Sjöberg 《Journal of Geodesy》1979,53(3):227-230
The method of Bjerhammar is studied in the continuous case for a sphere. By varying the kernel function, different types of
unknowns (u*) are obtained at the internal sphere (the Bjerhammar sphere). It is shown that a necessary condition for the existence of
u* is that the degree variances (σ
n
2
) of the observations are of an order less than n−2. According to Kaula’s rule this condition is not satisfied for the earth’s gravity anomaly field (σ
n
2
=n−1) but well for the geopotential (σ
n
2
=n−3). 相似文献
5.
A methodology for precise determination of the fundamental geodetic parameter w
0, the potential value of the Gauss–Listing geoid, as well as its time derivative 0, is presented. The method is based on: (1) ellipsoidal harmonic expansion of the external gravitational field of the Earth
to degree/order 360/360 (130 321 coefficients; http://www.uni-stuttgard.de/gi/research/ index.html projects) with respect
to the International Reference Ellipsoid WGD2000, at the GPS positioned stations; and (2) ellipsoidal free-air gravity reduction
of degree/order 360/360, based on orthometric heights of the GPS-positioned stations. The method has been numerically tested
for the data of three GPS campaigns of the Baltic Sea Level project (epochs 1990.8,1993.4 and 1997.4). New w
0 and 0 values (w
0=62 636 855.75 ± 0.21 m2/s2, 0=−0.0099±0.00079 m2/s2 per year, w
0/&γmacr;=6 379 781.502 m,0/&γmacr;=1.0 mm/year, and &γmacr;= −9.81802523 m2/s2) for the test region (Baltic Sea) were obtained. As by-products of the main study, the following were also determined: (1)
the high-resolution sea surface topography map for the Baltic Sea; (2) the most accurate regional geoid amongst four different
regional Gauss–Listing geoids currently proposed for the Baltic Sea; and (3) the difference between the national height datums
of countries around the Baltic Sea.
Received: 14 August 2000 / Accepted: 19 June 2001 相似文献
6.
C. C. Tscherning 《Journal of Geodesy》1999,73(6):332-336
A reproducing-kernel Hilbert space (RKHS) of functions harmonic in the set outside a sphere with radius R
0, having a reproducing kernel K
0(P,Q) is considered (P, Q, and later P
n
being points in the set of harmonicity). The degree variances of this kernel will be denoted σ0
n
.
The set of Riesz representers associated with the evaluation functionals (or gravity functionals) related to distinct points
P
n
,n = 1,…,N, on a two-dimensional surface surrounding the bounding sphere, will be linearly independent. These functions are used to
define a new N-dimensional RKHS with kernel (a
n
>0)
If the points all are located on a concentric sphere with radius R
1>R
0, and form an ε-net covering the sphere, and a
n
are suitable area elements (depending on N), then this kernel will converge towards an isotropic kernel with degree variances
Consequently, if K
N
(P,Q) is required to represent an isotropic covariance function of the Earth's gravity potential, COV(P,Q), σ0
n
can be selected so that σ
n
becomes equal to the empirical degree variances.
If the points are chosen at varying radial distances R
n
>R
0, then an anisotropic kernel, or equivalent covariance function representation, can be constructed. If the points are located
in a bounded region, the kernel may be used to modify the original kernel
Values of anisotropic covariance functions constructed based on these ideas are calculated, and some initial ideas are presented
on how to select the points P
n
.
Received: 24 September 1998 / Accepted: 10 March 1999 相似文献
7.
8.
A 2×2 arc-minute resolution geoid model, CARIB97, has been computed covering the Caribbean Sea. The geoid undulations refer
to the GRS-80 ellipsoid, centered at the ITRF94 (1996.0) origin. The geoid level is defined by adopting the gravity potential
on the geoid as W
0=62 636 856.88 m2/s2 and a gravity-mass constant of GM=3.986 004 418×1014 m3/s2. The geoid model was computed by applying high-frequency corrections to the Earth Gravity Model 1996 global geopotential
model in a remove-compute-restore procedure. The permanent tide system of CARIB97 is non-tidal. Comparison of CARIB97 geoid
heights to 31 GPS/tidal (ITRF94/local) benchmarks shows an average offset (h–H–N) of 51 cm, with an Root Mean Square (RMS) of 62 cm about the average. This represents an improvement over the use of a global
geoid model for the region. However, because the measured orthometric heights (H) refer to many differing tidal datums, these comparisons are biased by localized permanent ocean dynamic topography (PODT).
Therefore, we interpret the 51 cm as partially an estimate of the average PODT in the vicinity of the 31 island benchmarks.
On an island-by-island basis, CARIB97 now offers the ability to analyze local datum problems which were previously unrecognized
due to a lack of high-resolution geoid information in the area.
Received: 2 January 1998 / Accepted: 18 August 1998 相似文献
9.
E. Grafarend 《Journal of Geodesy》1970,44(1):41-49
Summary The probability to find an error vector in multiples of the Helmert-Maxwell-Boltzmann point error σ2 δij(δij Kronecker symbol) is calculated. It is found that the probability is for σ39%, for2 σ86% and for3 σ99% in two dimensions, for σ20%, for2 σ74% and for3 σ97% in three dimensions. The fundamental Maxwell-Boltzmann-distribution is tabulated0,02 (0,02) 4,50.
相似文献
10.
V. R. ?lander 《Journal of Geodesy》1954,28(4):329-342
Summary Consideration of the sources of error of the astronomical levelling appears to lead to an error or weight function of the
form (6), possibly in some cases the more general expression (7). The coefficients must in every single instance be empirically
determined from the material itself; here we primarily make use of the triangle closure errors in combination with the demand
that the mean error μ0 of the unit of weight be the same, independent of the size of the triangle.—Application of this procedure to the material
from Finland (255 stations of deviation of the vertical, combined into 337 triangles) can be regarded as a confirmation of
expression (6) with ε=±0″.30, κ=±0″.010/km, μ0=±6.7 cm (s
0=31.6 km). The coefficients are between themselves so similar, that their combined effect can hardly be distinguished from
a purely cubic function, which therefore was used as a base for the computation of the geoid.—On the other hand, bothDe Graaff-Hunter's material from Czecho-Slovakia [3] and that ofLitschauer from Austria [8] lead to purely quadratic functions of error, a result which can be interpreted so, that in these cases the
coefficient of the biquadratic interpolation term is so small, that it cannot be statistically demonstrated.
Zusammenfassung Eine Betrachtung der Fehlerquellen des astronomischen Nivellements scheint auf eine Fehler- oder Gewichtsfunktion von der Form (6), evtl. in gewissen F?llen auf den allgemeineren Ausdruck (7) zu führen. Die Koeffizienten sind auf empirischem Wege aus dem Material selbst zu bestimmen; hierbei bieten sich in erster Linie die Dreiecksschlussfehler in Kombination mit der Forderung, dass der mittlere Fehler μ0 der Gewichtseinheit von der Gr?sse des Dreiecks unabh?ngig sein muss. —Die Anwendung der Methode auf das finnische Material (255 Lotabweichungsstationen, zu 337 Dreiecken verbunden) kann als eine Best?tigung der Formel (6) gedeutet werden, mit ε=±0″.30, κ=±0″.010/km. μ0=±6.7 cm (s o=31.6 km). Die beiden Glieder sind einander so nahe gleich, dass ihre summe sich kaum von einer rein kubischen Funktion unterscheidet, und diese wurde daher der Geoidberechnung zu Grunde gelegt.—Dagegen führen sowohlDe Graaff-Hunters Material aus der Tscheckoslovakei [3] wie auchLitschauers Daten aus ?sterreich [8] auf rein quadratische Fehlerfunktionen, ein Ergebnis, das wohl dahin zu deuten ist, dass in diesen F?llen der Koeffizient des biquadratischen Interpolationsgliedes so klein ist, dass er sich nicht statistisch nachweisen l?sst.
Resumen El estudio de las causas de error en la nivelación astronómica conduce a un error de la forma (6) o de la forma más general (7) en la función de los pesos. En cada caso los coeficientes deben ser deducidos empíricamente de los resultados numéricos; se parte aquí del criterio de cierre de los triángulos y el del valor del error medio de la unidad de peso que debe ser independiente de las dimensiones del triángulo. La aplicación de este método a los resultados finlandeses (355 estaciones de desviaciones para 337 triángulos) puede considerase como una confirmación de la expresión (6) con . Los ceficientes son tan idénticos entre sí que su efecto combinado no puede distinguirse sino dificilmente de una función cúbica, que se ha utilizado para el cálculo del geoide. Los resultados del Dr.De Graaff-Hunter en Checoeslovaquia y de Mr.Litschauer en Austria dan los valores del 2° grado, que puede interpretarse diciendo que el término bicuadrático no es significativamente diferente de cero.
Résumé L'étude des causes d'erreur dans le nivellement astronomique amène à une erreur de la forme (6) ou de la forme plus générale (7) dans la fonction des poids. Dans chaque cas les coefficients doivent être tirés empiriquement des résultats numériques; on se base ici sur le criterium de fermeture de triangles et la valeur de l'erreur moyenne de l'unité de poids qui doit être indépendante des dimensions du triangle. L'application de cette méthode aux résultats Finlandais (255 stations de déviations pour 337 triangles) peut être considérée comme une confirmation de l'expression (6) avec ε=±0″.30 κ=±0″.010/km Les coefficients sont tellement identiques entre eux que leur effet combiné ne peut que difficilement se distinguer d'une fonction cubique que l'on a donc utilisée pour le calcul du géo?de. Les résultats du DrDe Graaff-Hunter en Tchécoslovaquie et deM. Litschauer en Autriche, donnent des valeurs du 2e degré, ce qui peut s'interpréter en disant que le terme biquadratique n'y est plus significantivement différent de zéro.
Sommario Lo studio delle cause di errore nella livellazione astronomica conduce ad un errore della forma (6) e della forma più generale (7) nella funzione dei pesi. In ciascun caso i coefficienti devono essere dedotti empiricamente dai risultati numerici; qui ci si basa sul criterio di chiusura dei triangoli ed il valore dell'errore medio dell'unità di peso che deve essere indipendente dalle dimensioni del triangolo. L'applicazione di questo metodo ai risultati finlandesi (255 stazioni di deviazione per 337 triangoli) può essere considerato come una conferma dell'espressione (6) con I coefficienti sono talmente identici tra di loro che il loro effetto combinato non può che difficilmente distinguersi dalla funzione cubica che é stata utilizzata per il calcolo del geoide. I risultati del Dr.De Graaff-Hunter in Cecoslovacchia e del Sig.Litschauer in Austria, forniscono dei valori del secondo ordine, ciò che può interpretarsi dicendo che il termine biquadratico non é significativamente differente dallo zero.相似文献
11.
Summary
Riemann polar/normal coordinates are the constituents to generate the oblique azimuthal projection of geodesic type, here applied to the reference ellipsoid of revolution (biaxial ellipsoid).Firstly we constitute a minimal atlas of the biaxial ellipsoid built on {ellipsoidal longitude, ellipsoidal latitude} and {metalongitude, metalatitude}. TheDarboux equations of a 1-dimensional submanifold (curve) in a 2-dimensional manifold (biaxial ellipsoid) are reviewed, in particular to represent geodetic curvature, geodetic torsion and normal curvature in terms of elements of the first and second fundamental form as well as theChristoffel symbols. The notion of ageodesic anda geodesic circle is given and illustrated by two examples. The system of twosecond order ordinary differential equations of ageodesic (Lagrange portrait) is presented in contrast to the system of twothird order ordinary differential equations of ageodesic circle (Proofs are collected inAppendix A andB). A precise definition of theRiemann mapping/mapping of geodesics into the local tangent space/tangent plane has been found.Secondly we computeRiemann polar/normal coordinates for the biaxial ellipsoid, both in theLagrange portrait (Legendre series) and in theHamilton portrait (Lie series).Thirdly we have succeeded in a detailed deformation analysis/Tissot distortion analysis of theRiemann mapping. The eigenvalues — the eigenvectors of the Cauchy-Green deformation tensor by means of ageneral eigenvalue-eigenvector problem have been computed inTable 3.1 andTable 3.2 (1, 2 = 1) illustrated inFigures 3.1, 3.2 and3.3. Table 3.3 contains the representation ofmaximum angular distortion of theRiemann mapping. Fourthly an elaborate global distortion analysis with respect toconformal Gau-Krüger, parallel Soldner andgeodesic Riemann coordinates based upon theAiry total deformation (energy) measure is presented in a corollary and numerically tested inTable 4.1. In a local strip [-l
E,l
E] = [-2°, +2°], [b
S,b
N] = [-2°, +2°]Riemann normal coordinates generate the smallest distortion, next are theparallel Soldner coordinates; the largest distortion by far is met by theconformal Gau-Krüger coordinates. Thus it can be concluded that for mapping of local areas of the biaxial ellipsoid surface the oblique azimuthal projection of geodesic type/Riemann polar/normal coordinates has to be favored with respect to others. 相似文献
12.
An operational algorithm for computation of terrain correction (or local gravity field modeling) based on application of closed-form solution of the Newton integral in terms of Cartesian coordinates in multi-cylindrical equal-area map projection of the reference ellipsoid is presented. Multi-cylindrical equal-area map projection of the reference ellipsoid has been derived and is described in detail for the first time. Ellipsoidal mass elements with various sizes on the surface of the reference ellipsoid are selected and the gravitational potential and vector of gravitational intensity (i.e. gravitational acceleration) of the mass elements are computed via numerical solution of the Newton integral in terms of geodetic coordinates {,,h}. Four base- edge points of the ellipsoidal mass elements are transformed into a multi-cylindrical equal-area map projection surface to build Cartesian mass elements by associating the height of the corresponding ellipsoidal mass elements to the transformed area elements. Using the closed-form solution of the Newton integral in terms of Cartesian coordinates, the gravitational potential and vector of gravitational intensity of the transformed Cartesian mass elements are computed and compared with those of the numerical solution of the Newton integral for the ellipsoidal mass elements in terms of geodetic coordinates. Numerical tests indicate that the difference between the two computations, i.e. numerical solution of the Newton integral for ellipsoidal mass elements in terms of geodetic coordinates and closed-form solution of the Newton integral in terms of Cartesian coordinates, in a multi-cylindrical equal-area map projection, is less than 1.6×10–8 m2/s2 for a mass element with a cross section area of 10×10 m and a height of 10,000 m. For a mass element with a cross section area of 1×1 km and a height of 10,000 m the difference is less than 1.5×10–4m2/s2. Since 1.5× 10–4 m2/s2 is equivalent to 1.5×10–5m in the vertical direction, it can be concluded that a method for terrain correction (or local gravity field modeling) based on closed-form solution of the Newton integral in terms of Cartesian coordinates of a multi-cylindrical equal-area map projection of the reference ellipsoid has been developed which has the accuracy of terrain correction (or local gravity field modeling) based on the Newton integral in terms of ellipsoidal coordinates.Acknowledgments. This research has been financially supported by the University of Tehran based on grant number 621/4/859. This support is gratefully acknowledged. The authors are also grateful for the comments and corrections made to the initial version of the paper by Dr. S. Petrovic from GFZ Potsdam and the other two anonymous reviewers. Their comments helped to improve the structure of the paper significantly. 相似文献
13.
Using a Love number formalism, the elastic deformations of the mantle and the mass redistribution gravitational potential within the Earth induced by the fluid pressure acting at the core–mantle boundary (CMB) are computed. This pressure field changes at a decadal time scale and may be estimated from observations of the surface magnetic field and its secular variation. First, using a spherical harmonic expansion, the poloidal and toroidal part of the fluid velocity field at the CMB for the last 40 years is computed, under the hypothesis of tangential geostrophy. Then the associated geostrophic pressure, whose order of magnitude is about 1000 Pa, is computed. The surface topography induced by this pressure field is computed using Love numbers, and is a few millimetres. The mass redistribution gravitational potential induced by these deformations and, in particular, the zonal components of the related surface gravitational potential perturbation (J2, J3 and J4 coefficients), are calculated. Overall perturbations for the J2 coefficient of about 10–10, for J3 of about 10–11 and for J4 are found of about 0.3×10–11. Finally, these theoretical results are compared with recent observations of the decadal variation of J2 from satellite laser ranging. Results concerning J2 can be described as follows: first, they are one order of magnitude too small to explain the observed decadal variation of J2 and, second, they show a significant linear trend over the last 40 years, whose rate of decrease amounts to 7% of the observed value. 相似文献
14.
Based on the gravity field models EGM96 and EIGEN-GL04C, the Earth＇s time-dependent principal moments of inertia A, B, C are obtained, and the variable rotation of the Earth is determined. Numerical results show that A, B, and C have increasing tendencies; the tilt of the rotation axis increases 2.1×10^ 8 mas/yr; the third component of the rotational angular velocity, ω3 , has a decrease of 1.0×10^ 22 rad/s^2, which is around 23% of the present observed value. Studies show in detail that both 0 and ω3 experience complex fluctuations at various time scales due to the variations of A, B and C. 相似文献
15.
Milan Burša Steve Kenyon Jan Kouba Zdislav Šíma Viliam Vatrt Vojtěch Vítek Marie Vojtíšková 《Journal of Geodesy》2007,81(2):103-110
The TOPEX/Poseidon (T/P) satellite alti- meter mission marked a new era in determining the geopotential constant W
0. On the basis of T/P data during 1993–2003 (cycles 11–414), long-term variations in W
0 have been investigated. The rounded value W
0 = 62636856.0 ± 0.5) m
2
s
−2 has already been adopted by the International Astronomical Union for the definition of the constant L
G
= W
0/c
2 = 6.969290134 × 10−10 (where c is the speed of light), which is required for the realization of the relativistic atomic time scale. The constant L
G
, based on the above value of W
0, is also included in the 2003 International Earth Rotation and Reference Frames Service conventions. It has also been suggested
that W
0 is used to specify a global vertical reference system (GVRS). W
0 ensures the consistency with the International Terrestrial Reference System, i.e. after adopting W
0, along with the geocentric gravitational constant (GM), the Earth’s rotational velocity (ω) and the second zonal geopotential
coefficient (J
2) as primary constants (parameters), then the ellipsoidal parameters (a,α) can be computed and adopted as derived parameters. The scale of the International Terrestrial Reference Frame 2000 (ITRF2000)
has also been specified with the use of W
0 to be consistent with the geocentric coordinate time. As an example of using W
0 for a GVRS realization, the geopotential difference between the adopted W
0 and the geopotential at the Rimouski tide-gauge point, specifying the North American Vertical Datum 1988 (NAVD88), has been
estimated. 相似文献
16.
N. R. Patel V. K. Dadhwal S. K. Saha 《Journal of the Indian Society of Remote Sensing》2011,39(3):383-391
The present study investigates the characteristics of CO2 exchange (photosynthesis and respiration) over agricultural site dominated by wheat crop and their relationship with ecosystem
parameters derived from MODIS. Eddy covariance measurement of CO2 and H2O exchanges was carried out at 10 Hz interval and fluxes of CO2 were computed at half-hourly time steps. The net ecosystem exchange (NEE) was partitioned into gross primary productivity
(GPP) and ecosystem respiration (R
e) by taking difference between day-time NEE and respiration. Time-series of daily reflectance and surface temperature products
at varying resolution (250–1000 m) were used to derive ecosystem variables (EVI, NDVI, LST). Diurnal pattern in Net ecosystem
exchange reveals negative NEE during day-time representing CO2 uptake and positive during night as release of CO2. The amplitude of the diurnal variation in NEE increased as LAI crop growth advances and reached its peak around the anthesis
stage. The mid-day uptake during this stage was around 1.15 mg CO2 m−2 s−1 and night-time release was around 0.15 mg CO2 m−2 s−1. Linear and non-linear least square regression procedures were employed to develop phenomenological models and empirical
fits between flux tower based GPP and NEE with satellite derived variables and environmental parameters. Enhanced vegetation
index was found significantly related to both GPP and NEE. However, NDVI showed little less significant relationship with
both GPP and NEE. Furthemore, temperature-greenness (TG) model combining scaled EVI and LST was parameterized to estimate
daily GPP over dominantly wheat crop site. (R
2 = 0.77). Multi-variate analysis shows that inclusion of LST or air temperature with EVI marginally improves variance explained
in daily NEE and GPP. 相似文献
17.
E.W. Grafarend 《Journal of Geodesy》2005,78(10):594-615
Harmonic maps are generated as a certain class of optimal map projections. For instance, if the distortion energy over a meridian strip of the International Reference Ellipsoid is minimized, we are led to the Laplace–Beltrami vector-valued partial differential equation. Harmonic functions x(L,B), y(L,B) given as functions of ellipsoidal surface parameters of Gauss ellipsoidal longitude L and Gauss ellipsoidal latitude B, as well as x(,q), y(,q) given as functions of relative isometric longitude =L–L0 and relative isometric latitude q=Q–Q0 gauged to a vector-valued boundary condition of special symmetry are constructed. The easting and northing {x(b,),y(b,)} of the new harmonic map is then given. Distortion energy analysis of the new harmonic map is presented, as well as case studies for (1) B[–40°,+40°], L[–31°,+49°], B0= ±30°, L0=9° and (2) B[46°,56°], L{[4.5°, 7.5°]; [7.5°, 10.5°]; [10.5°,13.5°]; [13.5°,16.5°]}, B0= 51°, L0 {6°,9°,12°,15°}. 相似文献
18.
《International Journal of Digital Earth》2013,6(7):687-700
ABSTRACTDigitizing the land surface temperature (Ts) and surface soil moisture (mv) is essential for developing the intelligent Digital Earth. Here, we developed a two parameter physical-based passive microwave remote sensing model for jointly retrieving Ts and mv using the dual-polarized Tb of Aqua satellite advanced microwave scanning radiometer (AMSR-E) C-band (6.9 GHz) based on the simplified radiative transfer equation. Validation using in situ Ts and mv in southern China showed the average root mean square errors (RMSE) of Ts and mv retrievals reach 2.42 K (R2 = 0.61, n = 351) and 0.025 g cm?3 (R2 = 0.68, n = 663), respectively. The results were also validated using global in situ Ts (n = 2362) and mv (n = 1657) of International Soil Moisture Network. The corresponding RMSE are 3.44 k (R2 = 0.86) and 0.039 g cm?3 (R2 = 0.83), respectively. The monthly variations of model-derived Ts and mv are highly consistent with those of the Moderate Resolution Imaging Spectroradiometer Ts (R2 = 0.57; RMSE = 2.91 k) and ECV_SM mv (R2 = 0.51; RMSE = 0.045 g cm?3), respectively. Overall, this paper indicates an effective way to jointly modeling Ts and mv using passive microwave remote sensing. 相似文献
19.
The six-hourly values of the atmospheric angular momentum (AAM) functions computed by the U.S. National Meteorological Center (NMC) were used to estimate the effects of the atmospheric tides on the Earth's rotation. Variations of the equatorial components
1 and
2 of the AAM have periods close to gravitational tidesP
1 andK
1.The amplitudes of the detected variations in
1 and
2 functions have been found to be much larger than the theoretical ones, the reason of this amplification remains unexplained. According to theoretical formulations, these waves can be expressed only as retrograde motions. Because of frame effects, there is a correspondance between diurnal retrograde polar motion and precession-nutations and the atmospheric effect on polar motion cannot be detected from observations.The second part of this paper deals the effects of atmospheric tides in Earth rotation. High-frequency UT1 variations have been derived from VLBI and GPS techniques during the SEARCH'92 campaign (Study ofEarth-AtmosphereRapidCHanges) (Dickey et al. 1994). They have been compared to values derived by Ray et al. (1994) from global ocean tide model. The results obtained in the present paper show the existence of variations of thermal origin with an amplitude of about 1µs in Universal Time UT1. The agreement between observed and theoretical values is better when the determined thermal atmospheric tides are taken into account.Oceanic tidal signal explains a large part (60% of the signal variance) of the diurnal and sub-diurnal variations. Our results show that only a small part of the residuals (5%) accounts for the atmospheric tidal effects. The residual signal remains unexplained; it might be due to mismodelization of oceanic or atmospheric tides or effect of other geophysical phenomena. 相似文献
20.
The regularized solution of the external sphericalStokes boundary value problem as being used for computations of geoid undulations and deflections of the vertical is based upon theGreen functions S
1(0, 0, , ) ofBox 0.1 (R = R
0) andV
1(0, 0, , ) ofBox 0.2 (R = R
0) which depend on theevaluation point {0, 0} S
R0
2
and thesampling point {, } S
R0
2
ofgravity anomalies
(, ) with respect to a normal gravitational field of typegm/R (free air anomaly). If the evaluation point is taken as the meta-north pole of theStokes reference sphere S
R0
2
, theStokes function, and theVening-Meinesz function, respectively, takes the formS() ofBox 0.1, andV
2() ofBox 0.2, respectively, as soon as we introduce {meta-longitude (azimuth), meta-colatitude (spherical distance)}, namely {A, } ofBox 0.5. In order to deriveStokes functions andVening-Meinesz functions as well as their integrals, theStokes andVening-Meinesz functionals, in aconvolutive form we map the sampling point {, } onto the tangent plane T0S
R0
2
at {0, 0} by means ofoblique map projections of type(i) equidistant (Riemann polar/normal coordinates),(ii) conformal and(iii) equiareal.Box 2.1.–2.4. andBox 3.1.– 3.4. are collections of the rigorously transformedconvolutive Stokes functions andStokes integrals andconvolutive Vening-Meinesz functions andVening-Meinesz integrals. The graphs of the correspondingStokes functions S
2(),S
3(r),,S
6(r) as well as the correspondingStokes-Helmert functions H
2(),H
3(r),,H
6(r) are given byFigure 4.1–4.5. In contrast, the graphs ofFigure 4.6–4.10 illustrate the correspondingVening-Meinesz functions V
2(),V
3(r),,V
6(r) as well as the correspondingVening-Meinesz-Helmert functions Q
2(),Q
3(r),,Q
6(r). The difference between theStokes functions / Vening-Meinesz functions andtheir first term (only used in the Flat Fourier Transforms of type FAST and FASZ), namelyS
2() – (sin /2)–1,S
3(r) – (sinr/2R
0)–1,,S
6(r) – 2R
0/r andV
2() + (cos /2)/2(sin2 /2),V
3(r) + (cosr/2R
0)/2(sin2
r/2R
0),,
illustrate the systematic errors in theflat Stokes function 2/ or flatVening-Meinesz function –2/2. The newly derivedStokes functions S
3(r),,S
6(r) ofBox 2.1–2.3, ofStokes integrals ofBox 2.4, as well asVening-Meinesz functionsV
3(r),,V
6(r) ofBox 3.1–3.3, ofVening-Meinesz integrals ofBox 3.4 — all of convolutive type — pave the way for the rigorousFast Fourier Transform and the rigorousWavelet Transform of theStokes integral / theVening-Meinesz integral of type equidistant, conformal and equiareal. 相似文献