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1.
Ebong  M. B. 《Journal of Geodesy》1986,60(4):279-287
Computer values of the mean square error of the Nigerian Geodetic Levelling based on the model xij = ρij/R ij 1/2 showed a uniformity between the lines of levelling, of a nature which suggest that xij = ρij/R ij 1/2 is a good fit for the data. Values of the mean square error of the levelling based on wij = ρij/Rij showed such significant variation between the lines that this alternative model does not seem to be supported by the data used in this study.  相似文献   

2.
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   

3.
The integral formulas of the associated Legendre functions   总被引：1，自引：0，他引：1  
A new kind of integral formulas for ${\bar{P}_{n,m} (x)}$ is derived from the addition theorem about the Legendre Functions when n ? m is an even number. Based on the newly introduced integral formulas, the fully normalized associated Legendre functions can be directly computed without using any recursion methods that currently are often used in the computations. In addition, some arithmetic examples are computed with the increasing degree recursion and the integral methods introduced in the paper respectively, in order to compare the precisions and run-times of these two methods in computing the fully normalized associated Legendre functions. The results indicate that the precisions of the integral methods are almost consistent for variant x in computing ${\bar{P}_{n,m} (x)}$ , i.e., the precisions are independent of the choice of x on the interval [0,1]. In contrast, the precisions of the increasing degree recursion change with different values on the interval [0,1], particularly, when x tends to 1, the errors of computing ${\bar{P}_{n,m} (x)}$ by the increasing degree recursion become unacceptable when the degree becomes larger and larger. On the other hand, the integral methods cost more run-time than the increasing degree recursion. Hence, it is suggested that combinations of the integral method and the increasing degree recursion can be adopted, that is, the integral methods can be used as a replacement for the recursive initials when the recursion method become divergent.  相似文献   

4.
Torsion balance observations in spherical approximation may be expressed as second-order partial derivatives of the anomalous (gravity) potential,T, $$T_{13} = \frac{{\partial ^2 T}}{{\partial x_1 \partial x_3 }}, T_{23} = \frac{{\partial ^2 T}}{{\partial x_2 \partial x_3 }}, T_{12} = \frac{{\partial ^2 T}}{{\partial x_1 \partial x_2 }}, T_\Delta = \frac{{\partial ^2 T}}{{\partial x_1^2 }} - \frac{{\partial ^2 T}}{{\partial x_1^2 }},$$ wherex 1 ,x 2 andx 3 are local coordinates withx 1 “east”,x 2 “north” andx 3 “up.” Auto- and cross-covariances for these quantities derived from an isotropic covariance function for the anomalous potential will depend on the directions between the observation points. However, the expressions for the covariances may be derived in a simple manner from isotropic covariance functions of torsion balance measurements. These functions are obtained by transforming the torsion balance observations in the points to local (orthogonal) horizontal coordinate systems with first axes in the direction to the other observation point. If the azimuth of the direction from one point to the other point is a, then the result of this transformation may be obtained by rotating the vectors $$\left\{ \begin{gathered} T_{13} \hfill \\ T_{23} \hfill \\ \end{gathered} \right\}and\left\{ \begin{gathered} T_\Delta \hfill \\ 2T_{12} \hfill \\ \end{gathered} \right\}$$ the angles a?90° and 2 (a?90°) respectively. The reverse rotations applied on the 2×2 matrices of covariances of these quantities will produce all the direction dependent covariances of the original quantities.  相似文献   

5.
Twenty-seven-day variation caused by solar rotation is one of the main periodic effects of solar radiation influence on the ionosphere, and there have been many studies on this periodicity using peak electron density \(\mathrm{N_{m}F_{2}}\) and solar radio flux index F10.7. In this paper, the global electron content (GEC) and observation of Solar EUV Monitor (SEM) represent the whole ionosphere and solar EUV flux, respectively, to investigate the 27-day variation. The 27-day period components of indices \((\hbox {GEC}_{27}\), \(\hbox {SEM}_{27}\), \(\hbox {F10.7}_{27}\), \(\hbox {Ap}_{27})\) are obtained using Chebyshev band-pass filter. The comparison of regression results indicates that the index SEM has higher coherence than F10.7 with 27-day variation of the ionosphere. The regression coefficients of \(\hbox {SEM}_{27 }\) varied from 0.6 to 1.4 and the coefficients of \(\hbox {Ap}_{27}\) varied from \({-}\)0.6 to 0.3, which suggests that EUV radiation seasonal variations are the primary driver for the 27-day variations of the ionosphere for most periods. TEC map grid points on three meridians where IGS stations are dense are selected for regression, and the results show that the contribution of solar EUV radiation is positive at all geomagnetic latitudes and larger than geomagnetic activity in most latitudes. The contribution of geomagnetic activity is negative at high geomagnetic latitude, increasing with decreasing geomagnetic latitudes, and positive at low geomagnetic latitudes. The global structure of 27-day variation of ionosphere is presented and demonstrates that there are two zonal anomaly regions along with the geomagnetic latitudes lines and two peaks in the north of Southeast Asia and the Middle Pacific where \(\hbox {TEC}_{27}\) magnitude values are notably larger than elsewhere along zonal anomaly regions.  相似文献   

6.
The frequency stability and uncertainty of the latest generation of optical atomic clocks is now approaching the one part in \(10^{18}\) level. Comparisons between earthbound clocks at rest must account for the relativistic redshift of the clock frequencies, which is proportional to the corresponding gravity (gravitational plus centrifugal) potential difference. For contributions to international timescales, the relativistic redshift correction must be computed with respect to a conventional zero potential value in order to be consistent with the definition of Terrestrial Time. To benefit fully from the uncertainty of the optical clocks, the gravity potential must be determined with an accuracy of about \(0.1\,\hbox {m}^{2}\,\hbox {s}^{-2}\), equivalent to about 0.01 m in height. This contribution focuses on the static part of the gravity field, assuming that temporal variations are accounted for separately by appropriate reductions. Two geodetic approaches are investigated for the derivation of gravity potential values: geometric levelling and the Global Navigation Satellite Systems (GNSS)/geoid approach. Geometric levelling gives potential differences with millimetre uncertainty over shorter distances (several kilometres), but is susceptible to systematic errors at the decimetre level over large distances. The GNSS/geoid approach gives absolute gravity potential values, but with an uncertainty corresponding to about 2 cm in height. For large distances, the GNSS/geoid approach should therefore be better than geometric levelling. This is demonstrated by the results from practical investigations related to three clock sites in Germany and one in France. The estimated uncertainty for the relativistic redshift correction at each site is about \(2 \times 10^{-18}\).  相似文献   

7.
The conventional expansions of the gravity gradients in the local north-oriented reference frame have a complicated form, depending on the first- and second-order derivatives of the associated Legendre functions of the colatitude and containing factors which tend to infinity when approaching the poles. In the present paper, the general term of each of these series is transformed to a product of a geopotential coefficient and a sum of several adjacent Legendre functions of the colatitude multiplied by a function of the longitude. These transformations are performed on the basis of relations between the Legendre functions and their derivatives published by Ilk (1983). The second-order geopotential derivatives corresponding to the local orbital reference frame are presented as linear functions of the north-oriented gravity gradients. The new expansions for the latter are substituted into these functions. As a result, the orbital derivatives are also presented as series depending on the geopotential coefficients multiplied by sums of the Legendre functions whose coefficients depend on the longitude and the satellite track azimuth at an observation point. The derived expansions of the observables can be applied for constructing a geopotential model from the GOCE mission data by the time-wise and space-wise approaches. The numerical experiments demonstrate the correctness of the analytical formulas.An erratum to this article can be found at  相似文献   

8.
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   

9.
We present results from a new vertical deflection (VD) traverse observed in Perth, Western Australia, which is the first of its kind in the Southern Hemisphere. A digital astrogeodetic QDaedalus instrument was deployed to measure VDs with \({\sim }\)0.2\(''\) precision at 39 benchmarks with a \({{\sim }}1~\hbox {km}\) spacing. For the conversion of VDs to quasigeoid height differences, the method of astronomical–topographical levelling was applied, based on topographical information from the Shuttle Radar Topography Mission. The astronomical quasigeoid heights are in 20–30 mm (RMS) agreement with three independent gravimetric quasigeoid models, and the astrogeodetic VDs agree to 0.2–0.3\(''\) (north–south) and 0.6–0.9\(''\) (east–west) RMS. Tilt-like biases of \({\sim }1\,\,\hbox {mm}\) over \({\sim }1\,\,\hbox {km}\) are present for all quasigeoid models within \({\sim }20\,\,\hbox {km}\) of the coastline, suggesting inconsistencies in the coastal zone gravity data. The VD campaign in Perth was designed as a low-cost effort, possibly allowing replication in other Southern Hemisphere countries (e.g., Asia, Africa, South America and Antarctica), where VD data are particularly scarce.  相似文献   

10.
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   

11.
We present new insights on the time-averaged surface velocities, convergence and extension rates along arc-normal transects in Kumaon, Garhwal and Kashmir–Himachal regions in the Indian Himalaya from 13 years of high-precision Global Positioning System (GPS) time series (1995–2008) derived from GPS data at 14 GPS permanent and 42 campaign stations between $29.5{-}35^{\circ }\hbox {N}$ and $76{-}81^{\circ }\hbox {E}$ . The GPS surface horizontal velocities vary significantly from the Higher to Lesser Himalaya and are of the order of 30 to 48 mm/year NE in ITRF 2005 reference frame, and 17 to 2 mm/year SW in an India fixed reference frame indicating that this region is accommodating less than 2 cm/year of the India–Eurasia plate motion ( ${\sim }4~\hbox {cm/year}$ ). The total arc-normal shortening varies between ${\sim }10{-}14~\hbox {mm/year}$ along the different transects of the northwest Himalayan wedge, between the Indo-Tsangpo suture to the north and the Indo-Gangetic foreland to the south indicating high strain accumulation in the Himalayan wedge. This convergence is being accommodated differentially along the arc-normal transects; ${\sim } 5{-}10~\hbox {mm/year}$ in Lesser Himalaya and 3–4 mm/year in Higher Himalaya south of South Tibetan Detachment. Most of the convergence in the Lesser Himalaya of Garhwal and Kumaon is being accommodated just south of the Main Central Thrust fault trace, indicating high strain accumulation in this region which is also consistent with the high seismic activity in this region. In addition, for the first time an arc-normal extension of ${\sim }6~\hbox {mm/year}$ has also been observed in the Tethyan Himalaya of Kumaon. Inverse modeling of GPS-derived surface deformation rates in Garhwal and Kumaon Himalaya using a single dislocation indicate that the Main Himalayan Thrust is locked from the surface to a depth of ${\sim }15{-}20~\hbox {km}$ over a width of 110 km with associated slip rate of ${\sim }16{-}18~\hbox {mm/year}$ . These results indicate that the arc-normal rates in the Northwest Himalaya have a complex deformation pattern involving both convergence and extension, and rigorous seismo-tectonic models in the Himalaya are necessary to account for this pattern. In addition, the results also gave an estimate of co-seismic and post-seismic motion associated with the 1999 Chamoli earthquake, which is modeled to derive the slip and geometry of the rupture plane.  相似文献   

12.
M-estimation with probabilistic models of geodetic observations   总被引：1，自引：1，他引：0  
The paper concerns \(M\) -estimation with probabilistic models of geodetic observations that is called \(M_{\mathcal {P}}\) estimation. The special attention is paid to \(M_{\mathcal {P}}\) estimation that includes the asymmetry and the excess kurtosis, which are basic anomalies of empiric distributions of errors of geodetic or astrometric observations (in comparison to the Gaussian errors). It is assumed that the influence function of \(M_{\mathcal {P}}\) estimation is equal to the differential equation that defines the system of the Pearson distributions. The central moments \(\mu _{k},\, k=2,3,4\) , are the parameters of that system and thus, they are also the parameters of the chosen influence function. The \(M_{\mathcal {P}}\) estimation that includes the Pearson type IV and VII distributions ( \(M_{\mathrm{PD(l)}}\) method) is analyzed in great detail from a theoretical point of view as well as by applying numerical tests. The chosen distributions are leptokurtic with asymmetry which refers to the general characteristic of empirical distributions. Considering \(M\) -estimation with probabilistic models, the Gram–Charlier series are also applied to approximate the models in question ( \(M_{\mathrm{G-C}}\) method). The paper shows that \(M_{\mathcal {P}}\) estimation with the application of probabilistic models belongs to the class of robust estimations; \(M_{\mathrm{PD(l)}}\) method is especially effective in that case. It is suggested that even in the absence of significant anomalies the method in question should be regarded as robust against gross errors while its robustness is controlled by the pseudo-kurtosis.  相似文献   

13.
Tropical cyclones when on land create havoc, but over the oceans they can trigger a very strong biological response, giving rise to phytoplankton blooms. The Super Cyclone (TC) 05B that occurred during October 25–29, 1999, in the Bay of Bengal over the tropical Indian ocean was one of the most significant tropical cyclones on record to affect India, with maximum winds of 240 km/h, and the worst since 1971. Using satellite data, it is found that this tropical super cyclone helped spawn a notable mesoscale phytoplankton bloom in the domain (17 $^{circ}$–20$^{circ} hbox{N}$; 87$^{circ}$–90 $^{circ} hbox{E}$), which persisted for over a month. The bloom spanned 440 km zonally and 330 km meridonally, enhanced the chlorophyll-$a$ concentrations to a maximum of 10 $hbox{mg/m}^{3}$ and the net primary productivity by 200%. Furthermore, a cyclonic eddy over the bloom region is revealed from an ocean general circulation model simulation, helping the bloom to last for over month.   相似文献   

14.
We can map zenith wet delays onto precipitable water with a conversion factor, but in order to calculate the exact conversion factor, we must precisely calculate its key variable $T_\mathrm{m}$ . Yao et al. (J Geod 86:1125–1135, 2012. doi:10.1007/s00190-012-0568-1) established the first generation of global $T_\mathrm{m}$ model (GTm-I) with ground-based radiosonde data, but due to the lack of radiosonde data at sea, the model appears to be abnormal in some areas. Given that sea surface temperature varies less than that on land, and the GPT model and the Bevis $T_\mathrm{m}$ – $T_\mathrm{s}$ relationship are accurate enough to describe the surface temperature and $T_\mathrm{m}$ , this paper capitalizes on the GPT model and the Bevis $T_\mathrm{m}$ – $T_\mathrm{s}$ relationship to provide simulated $T_\mathrm{m}$ at sea, as a compensation for the lack of data. Combined with the $T_\mathrm{m}$ from radiosonde data, we recalculated the GTm model coefficients. The results show that this method not only improves the accuracy of the GTm model significantly at sea but also improves that on land, making the GTm model more stable and practically applicable.  相似文献   

15.
The well known least squares collocation model (I) $$\ell = Ax + \left[ {\begin{array}{*{20}c} O \\ I \\ \end{array} } \right]^T \left[ {\begin{array}{*{20}c} s \\ {s' + n} \\ \end{array} } \right]$$ is compared with the model (II) $$\ell = Ax + \left[ {\begin{array}{*{20}c} R \\ I \\ \end{array} } \right]^T \left[ {\begin{array}{*{20}c} s \\ n \\ \end{array} } \right]$$ The basic differences of these two models in the framework of physical geodesy are pointed out by analyzing the validity of the equation $$s' = Rs$$ that transforms one model into the other, for different cases. For clarification purposes least squares filtering, prediction and collocation are discussed separately. In filtering problems the coefficient matrix R becomes the unit matrix and by this the two models become identical. For prediction and collocation problems the relation s′=Rs is only fulfilled in the global limit where s becomes either a continuous function on the earth or an intinite set of spherical harmonic coefficients. Applying Model (II), we see that for any finite dimension of s the operator equations of physical geodesy are approximated by a finite matrix relation whereas in Model (I) the operator equations are applied in their correct form on a continuous, approximate function \(\tilde s\) .  相似文献   

16.
Computation of broadcast ephemerides is a fundamental task in satellite navigation and positioning. The GPS constellation is composed of medium-earth-orbit (MEO) satellites, and therefore can employ a uniform parameter set to produce broadcast ephemerides. However, other navigation satellite systems such as Compass and IRNSS may include a mixture of inclined-geosynchronous-orbit (IGSO), geostationary-earth-orbit (GEO) and MEO satellites, requiring different parameter sets for each type of orbit. We analyze the variational characteristics of satellite ephemerides with respect to orbital elements; then present a method to design an optimal parameter set for broadcast ephemerides, and derive the parameter sets for IGSO, GEO, and MEO satellites. The computational complexities of the user algorithms for the optimal parameter sets are equivalent to that of the standard GPS user algorithm. Simulation and statistical analyses indicate that the optimal parameter set is $ \left\{ {\sqrt {A_{0} } ,e_{0} ,i_{0} ,\Upomega_{0} ,M_{0} ,\omega_{0} ,\dot{\Upomega },\dot{u},\dot{i},C_{\Upomega c3} ,C_{\Upomega s3} ,C_{uc2} ,C_{us2} ,C_{rc2} ,C_{rs2} } \right\} $ for IGSO and GEO satellites, and $ \left\{ {\sqrt {A_{0} } ,e_{0} ,i_{0} ,\Upomega_{0} ,M_{0} ,\omega_{0} ,\dot{\Upomega },\dot{u},\dot{i},C_{uc2} ,C_{us2} ,C_{rc2} ,C_{rs2} ,C_{ic2} ,C_{is2} } \right\} $ for MEO satellites.  相似文献   

17.
On weighted total least-squares for geodetic transformations   总被引：5，自引：4，他引：1  
  相似文献   

18.
Large-scale mass redistribution in the terrestrial water storage (TWS) leads to changes in the low-degree spherical harmonic coefficients of the Earth’s surface mass density field. Studying these low-degree fluctuations is an important task that contributes to our understanding of continental hydrology. In this study, we use global GNSS measurements of vertical and horizontal crustal displacements that we correct for atmospheric and oceanic effects, and use a set of modified basis functions similar to Clarke et al. (Geophys J Int 171:1–10, 2007) to perform an inversion of the corrected measurements in order to recover changes in the coefficients of degree-0 (hydrological mass change), degree-1 (centre of mass shift) and degree-2 (flattening of the Earth) caused by variations in the TWS over the period January 2003–January 2015. We infer from the GNSS-derived degree-0 estimate an annual variation in total continental water mass with an amplitude of \((3.49 \pm 0.19) \times 10^{3}\) Gt and a phase of \(70^{\circ } \pm 3^{\circ }\) (implying a peak in early March), in excellent agreement with corresponding values derived from the Global Land Data Assimilation System (GLDAS) water storage model that amount to \((3.39 \pm 0.10) \times 10^{3}\) Gt and \(71^{\circ } \pm 2^{\circ }\), respectively. The degree-1 coefficients we recover from GNSS predict annual geocentre motion (i.e. the offset change between the centre of common mass and the centre of figure) caused by changes in TWS with amplitudes of \(0.69 \pm 0.07\) mm for GX, \(1.31 \pm 0.08\) mm for GY and \(2.60 \pm 0.13\) mm for GZ. These values agree with GLDAS and estimates obtained from the combination of GRACE and the output of an ocean model using the approach of Swenson et al. (J Geophys Res 113(B8), 2008) at the level of about 0.5, 0.3 and 0.9 mm for GX, GY and GZ, respectively. Corresponding degree-1 coefficients from SLR, however, generally show higher variability and predict larger amplitudes for GX and GZ. The results we obtain for the degree-2 coefficients from GNSS are slightly mixed, and the level of agreement with the other sources heavily depends on the individual coefficient being investigated. The best agreement is observed for \(T_{20}^C\) and \(T_{22}^S\), which contain the most prominent annual signals among the degree-2 coefficients, with amplitudes amounting to \((5.47 \pm 0.44) \times 10^{-3}\) and \((4.52 \pm 0.31) \times 10^{-3}\) m of equivalent water height (EWH), respectively, as inferred from GNSS. Corresponding agreement with values from SLR and GRACE is at the level of or better than \(0.4 \times 10^{-3}\) and \(0.9 \times 10^{-3}\) m of EWH for \(T_{20}^C\) and \(T_{22}^S\), respectively, while for both coefficients, GLDAS predicts smaller amplitudes. Somewhat lower agreement is obtained for the order-1 coefficients, \(T_{21}^C\) and \(T_{21}^S\), while our GNSS inversion seems unable to reliably recover \(T_{22}^C\). For all the coefficients we consider, the GNSS-derived estimates from the modified inversion approach are more consistent with the solutions from the other sources than corresponding estimates obtained from an unconstrained standard inversion.  相似文献   

19.
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   

20.
Well credited and widely used ionospheric models, such as the International Reference Ionosphere or NeQuick, describe the variation of the electron density with height by means of a piecewise profile tied to the F2-peak parameters: the electron density, $N_m \mathrm{F2}$ N m F 2 , and the height, $h_m \mathrm{F2}$ h m F 2 . Accurate values of these parameters are crucial for retrieving reliable electron density estimations from those models. When direct measurements of these parameters are not available, the models compute the parameters using the so-called ITU-R database, which was established in the early 1960s. This paper presents a technique aimed at routinely updating the ITU-R database using radio occultation electron density profiles derived from GPS measurements gathered from low Earth orbit satellites. Before being used, these radio occultation profiles are validated by fitting to them an electron density model. A re-weighted Least Squares algorithm is used for down-weighting unreliable measurements (occasionally, entire profiles) and to retrieve $N_m \mathrm{F2}$ N m F 2 and $h_m \mathrm{F2}$ h m F 2 values—together with their error estimates—from the profiles. These values are used to monthly update the database, which consists of two sets of ITU-R-like coefficients that could easily be implemented in the IRI or NeQuick models. The technique was tested with radio occultation electron density profiles that are delivered to the community by the COSMIC/FORMOSAT-3 mission team. Tests were performed for solstices and equinoxes seasons in high and low-solar activity conditions. The global mean error of the resulting maps—estimated by the Least Squares technique—is between $0.5\times 10^{10}$ 0.5 × 10 10 and $3.6\times 10^{10}$ 3.6 × 10 10 elec/m $^{-3}$ ? 3 for the F2-peak electron density (which is equivalent to 7 % of the value of the estimated parameter) and from 2.0 to 5.6 km for the height ( $\sim $ ～ 2 %).  相似文献   

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