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1.
One of the most basic and important tools in optimal spectral gravity field modelling is the method of Wiener filtering.
Originally developed for applications in analogue signal analysis and communication engineering, Wiener filtering has become
a standard linear estimation technique of modern operational geodesy, either as an independent practical tool for data de-noising
in the frequency domain or as an integral component of a more general signal estimation methodology (input–output systems
theory). Its theoretical framework is based on the Wiener–Kolmogorov linear prediction theory for stationary random fields
in the presence of additive external noise, and thus it is closely related to the (more familiar to geodesists) method of
least-squares collocation with random observation errors. The main drawback of Wiener filtering that makes its use in many
geodetic applications problematic stems from the stationarity assumption for both the signal and the noise involved in the
approximation problem. A modified Wiener-type linear estimation filter is introduced that can be used with noisy data obtained
from an arbitrary deterministic field under the masking of non-stationary random observation errors. In addition, the sampling
resolution of the input data is explicitly taken into account within the estimation algorithm, resulting in a resolution-dependent
optimal noise filter. This provides a more insightful approach to spectral filtering techniques for noise reduction, since
the data resolution parameter has not been directly incorporated in previous formulations of frequency-domain estimation problems
for gravity field signals with discrete noisy data.
Received: 1 November 2000 / Accepted: 19 June 2001 相似文献
2.
Five separate polar motion series are examined in order to understand what portion of their variations at periods exceeding
several years represents true polar motion. The data since the development of space-geodetic techniques (by themselves insufficient
for study of long-period motion), and a variety of historical astrometric data sets, allow the following tentative conclusions:
retrograde long-period polar motion below about −0.2 cpy (cycles per year) in pre-space-geodetic data (pre-1976) is dominantly
noise. For 1976–1992, there is poor agreement between space-geodetic and astrometric series over the range −0.2 to +0.2 cpy,
demonstrating that classical astrometry lacked the precision to monitor polar motion in this frequency range. It is concluded
that all the pre-1976 astrometric polar motion data are likely to be dominated by noise at periods exceeding about 10 years.
The exception to this is possibly a linear trend found in some astrometric and space geodetic series. At frequencies above
prograde +0.2 cpy (periods shorter than about 5 years), historical astrometric data may be of sufficient quality for comparisons
with geophysical excitation time series. Even in the era of space geodesy, significant differences are found in long-period
variations in published polar motion time series.
Received: 27 March 2001 / Accepted: 15 October 2001 相似文献
3.
A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions 总被引:11,自引:7,他引:11
Spherical harmonic expansions form partial sums of fully normalised associated Legendre functions (ALFs). However, when evaluated
increasingly close to the poles, the ultra-high degree and order (e.g. 2700) ALFs range over thousands of orders of magnitude.
This causes existing recursion techniques for computing values of individual ALFs and their derivatives to fail. A common
solution in geodesy is to evaluate these expansions using Clenshaw's method, which does not compute individual ALFs or their
derivatives. Straightforward numerical principles govern the stability of this technique. Elementary algebra is employed to
illustrate how these principles are implemented in Clenshaw's method. It is also demonstrated how existing recursion algorithms
for computing ALFs and their first derivatives are easily modified to incorporate these same numerical principles. These modified
recursions yield scaled ALFs and first derivatives, which can then be combined using Horner's scheme to compute partial sums,
complete to degree and order 2700, for all latitudes (except at the poles for first derivatives). This exceeds any previously
published result. Numerical tests suggest that this new approach is at least as precise and efficient as Clenshaw's method.
However, the principal strength of the new techniques lies in their simplicity of formulation and implementation, since this
quality should simplify the task of extending the approach to other uses, such as spherical harmonic analysis.
Received: 30 June 2000 / Accepted: 12 June 2001 相似文献
4.
A global plate motion model is established based on the ITRF97 velocity fields and geological model NUVEL1. Sub-plate models
are estimated by using the velocity fields derived from 45 global positioning system (GPS) sites under the ITRF97 reference
frame in China. Comparisons between space geodesy and geological models are given. It is found that the Euler vector of the
AFRC–EURA pair has an obvious discrepancy between space geodetic and geological models. The motion patterns of tectonic blocks
in China predicted by GPS are consistent with those of geological data on the whole.
Received: 9 November 2000 / Accepted: 17 September 2001 相似文献
5.
Length-of-day (LOD) estimates from seven Global Positioning System (GPS) and three satellite laser ranging (SLR) analysis
centers were combined into an even-spaced time series for a 27-month period (1996–1998). This time series was compared to
the multi-technique Earth-orientation-parameter (EOP) combined solution (C04) derived at the Central Bureau of the International
Earth Rotation Service (IERS/CB). Due to inhomogeneities in the different series derived from the various techniques (time,
length, quality, and spatial resolution), the concept of a combined solution is justified. The noise behavior in LOD for different
techniques varies with frequency; the data series were divided into frequency windows after removing both biases and trends.
Different weight factors were assigned in each window, discriminating by technique, and produced one-technique combined solutions.
Finally, these one-technique combined solutions were combined to obtain the final multi-technique solution. The LOD combined
time series obtained by the present method based on the frequency windows combined series (FWCS) is very close to the IERS
C04 solution. It could be useful to generate a new LOD reference time series to be used in the study of high-frequency variations
of Earth rotation.
Received: 28 March 2000 / Accepted: 15 February 2001 相似文献
6.
The performance of the L-curve criterion and of the generalized cross-validation (GCV) method for the Tikhonov regularization
of the ill-conditioned normal equations associated with the determination of the gravity field from satellite gravity gradiometry
is investigated. Special attention is devoted to the computation of the corner point of the L-curve, to the numerically efficient
computation of the trace term in the GCV target function, and to the choice of the norm of the residuals, which is important
for the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) in the presence of colored observation noise. The
trace term in the GCV target function is estimated using an unbiased minimum-variance stochastic estimator. The performance
analysis is based on a simulation of gravity gradients along a 60-day repeat circular orbit and a gravity field recovery complete
up to degree and order 300. Randomized GCV yields the optimal regularization parameter in all the simulations if the colored
noise is properly taken into account. Moreover, it seems to be quite robust against the choice of the norm of the residuals.
It performs much better than the L-curve criterion, which always yields over-smooth solutions. The numerical costs for randomized
GCV are limited provided that a reasonable first guess of the regularization parameter can be found.
Received: 17 May 2001 / Accepted: 17 January 2002 相似文献
7.
A new method for calculating analytical solar radiation pressure models for GNSS spacecraft has been developed. The method
simulates the flux of light from the Sun using a pixel array. The method can cope with a high level of complexity in the spacecraft
structure and models effects due to reflected light. Models have been calculated and tested for the Russhar global navigation
satellite system GLONASS IIv spacecraft. Results are presented using numerical integration of the force model and long-arc
satellite laser ranging (SLR) analysis. The integrated trajectory differs from a precise orbit calculated using a network
of global tracking stations by circa 2 m root mean square over a 160 000-km arc. The observed − computed residuals for the
400-day SLR arc are circa 28 mm.
Received: 23 December 1999 / Accepted: 28 August 2000 相似文献
8.
E. W. Grafarend 《Journal of Geodesy》2001,75(7-8):363-390
In a comparison of the solution of the spherical horizontal and vertical boundary value problems of physical geodesy it is
aimed to construct downward continuation operators for vertical deflections (surface gradient of the incremental gravitational
potential) and for gravity disturbances (vertical derivative of the incremental gravitational potential) from points on the
Earth's topographic surface or of the three-dimensional (3-D) Euclidean space nearby down to the international reference sphere
(IRS). First the horizontal and vertical components of the gravity vector, namely spherical vertical deflections and spherical
gravity disturbances, are set up. Second, the horizontal and vertical boundary value problem in spherical gravity and geometry
space is considered. The incremental gravity vector is represented in terms of vector spherical harmonics. The solution of
horizontal spherical boundary problem in terms of the horizontal vector-valued Green function converts vertical deflections
given on the IRS to the incremental gravitational potential external in the 3-D Euclidean space. The horizontal Green functions
specialized to evaluation and source points on the IRS coincide with the Stokes kernel for vertical deflections. Third, the
vertical spherical boundary value problem is solved in terms of the vertical scalar-valued Green function. Fourth, the operators
for upward continuation of vertical deflections given on the IRS to vertical deflections in its external 3-D Euclidean space
are constructed. Fifth, the operators for upward continuation of incremental gravity given on the IRS to incremental gravity
to the external 3-D Euclidean space are generated. Finally, Meissl-type diagrams for upward continuation and regularized downward
continuation of horizontal and vertical gravity data, namely vertical deflection and incremental gravity, are produced.
Received: 10 May 2000 / Accepted: 26 February 2001 相似文献
9.
L. E. Sjöberg 《Journal of Geodesy》2000,74(2):255-268
The topographic potential and the direct topographic effect on the geoid are presented as surface integrals, and the direct
gravity effect is derived as a rigorous surface integral on the unit sphere. By Taylor-expanding the integrals at sea level
with respect to topographic elevation (H) the power series of the effects is derived to arbitrary orders. This study is primarily limited to terms of order H
2. The limitations of the various effects in the frequently used planar approximations are demonstrated. In contrast, it is
shown that the spherical approximation to power H
2 leads to a combined topographic effect on the geoid (direct plus indirect effect) proportional to H˜2 (where terms of degrees 0 and 1 are missing) of the order of several metres, while the combined topographic effect on the
height anomaly vanishes, implying that current frequent efforts to determine the direct effect to this order are not needed.
The last result is in total agreement with Bjerhammar's method in physical geodesy. It is shown that the most frequently applied
remove–restore technique of topographic masses in the application of Stokes' formula suffers from significant errors both
in the terrain correction C (representing the sum of the direct topographic effect on gravity anomaly and the effect of continuing the anomaly to sea
level) and in the term t (mainly representing the indirect effect on the geoidal or quasi-geoidal height).
Received: 18 August 1998 / Accepted: 4 October 1999 相似文献
10.
Station coordinates are combined with velocities estimated by space geodesy techniques to produce the International Terrestrial
Reference System. The input is sets of coordinates and velocities calculated by International Earth Rotation Service analysis
centers using space geodesy techniques. The working reference system of individual analysis centers is generally conventionally
defined. However, the implications of such processing can have an effect on the resulting combined set. The problem of datum
definition as a function of coordinate combinations is reviewed. In particular, the problem of minimum constraints is clearly
emphasized and the reference system effect is defined. The goal is to build a process that could be used generally to remove
uncertainties in the underlying coordinate system without disturbing the underlying information with additional unnecessary
information.
Received: 25 January 1999 / Accepted: 20 September 2000 相似文献
11.
R. Lehmann 《Journal of Geodesy》2000,74(3-4):327-334
The definition and connection of vertical datums in geodetic height networks is a fundamental problem in geodesy. Today,
the standard approach to solve it is based on the joint processing of terrestrial and satellite geodetic data. It is generalized
to cases where the coverage with terrestrial data may change from region to region, typically across coastlines. The principal
difficulty is that such problems, so-called altimetry–gravimetry boundary-value problems (AGPs), do not admit analytical solutions
such as Stokes' integral. A numerical solution strategy for the free-datum problem is presented. Analysis of AGPs in spherical
and constant radius approximation shows that two of them are mathematically well-posed problems, while the classical AGP-I
may be ill posed in special situations.
Received: 2 December 1998 / Accepted: 30 November 1999 相似文献
12.
The multiresolution character of collocation 总被引:3,自引:0,他引:3
C. Kotsakis 《Journal of Geodesy》2000,74(3-4):275-290
An interesting theoretical connection between the statistical (non-stochastic) collocation principle and the multiresolution/wavelet
framework of signal approximation is presented. The rapid developments in multiresolution analysis theory over the past few
years have provided very useful (theoretical and practical) tools for approximation and spectral studies of irregularly varying
signals, thus opening new possibilities for `non-stationary' gravity field modeling. It is demonstrated that the classic multiresolution
formalism according to Mallat's pioneering work lies at the very core of some of the general approximation principles traditionally
used in physical geodesy problems. In particular, it is shown that the use of a spatio-statistical (non-probabilistic) minimum
mean-square-error criterion for optimal linear estimation of deterministic signals, in conjunction with regularly gridded
data, always gives rise to a generalized multiresolution analysis in the Hilbert space L
2(R), under some mild constraints on the spatial covariance function and the power spectrum of the unknown field under consideration.
Using the theory and the actual approximation algorithms associated with statistical collocation, a new constructive framework
for building generalized multiresolution analyses in L
2(R) is presented, without the need for the usual dyadic restriction that exists in classic wavelet theory. The multiresolution and `non-stationary' aspects of the statistical collocation approximation
procedure are also discussed, and finally some conclusions and recommendations for future work are given.
Received: 26 January 1999 / Accepted: 16 August 1999 相似文献
13.
A formula for computing the gravity disturbance from the second radial derivative of the disturbing potential 总被引:6,自引:0,他引:6
J. Li 《Journal of Geodesy》2002,76(4):226-231
A formula for computing the gravity disturbance and gravity anomaly from the second radial derivative of the disturbing potential
is derived in detail using the basic differential equation with spherical approximation in physical geodesy and the modified
Poisson integral formula. The derived integral in the space domain, expressed by a spherical geometric quantity, is then converted
to a convolution form in the local planar rectangular coordinate system tangent to the geoid at the computing point, and the
corresponding spectral formulae of 1-D FFT and 2-D FFT are presented for numerical computation.
Received: 27 December 2000 / Accepted: 3 September 2001 相似文献
14.
The identification of mean semi-major axes (suitably defined) for satellite orbits to satisfy a variety of requirements for
geodesy, geophysics and oceanography, in terms of repeat orbits (with orbital resonances), is investigated. Various options
for the definition of semi-major axis, from the viewpoint of satellite dynamics, are described. Simple simulations of the
expected resonant changes in inclination are presented, and tools for the analysis of orbit resonances to extract certain
lumped harmonic coefficients of the geopotential (e.g. from the very precise CHAMP orbit) are resurrected. Finally, a preliminary
example of the 46th-order resonance analysis possible for CHAMP, based on the mean orbital elements produced by GFZ (GeoForschungs
Zentrum) for ephemeris prediction, is presented.
Received: 10 July 2001 / Accepted: 17 July 2002
Correspondence to: J. Klokočník at Ondřejov Observatory
Acknowledgements. We thank Prof. Dr. Ch. Reigber, Dr. P. Schwintzer, Dr. T. Gruber and Dr. R. K?nig from GFZ Potsdam for various consultations
and discussions, and for the CHAMP two-line mean elements. This investigation was performed under the aegis of CEDR (Center
for Earth's Dynamics Research, Prague-Ondřejov); it has been supported by project LN00A005 (provided by the Ministry of Education
of the Czech Republic) and by grant A 3004 of the Grant Agency of the Academy of Sciences of the Czech Republic. 相似文献
15.
Local geoid determination combining gravity disturbances and GPS/levelling: a case study in the Lake Nasser area, Aswan, Egypt 总被引:1,自引:0,他引:1
C. C. Tscherning Awar Radwan A. A. Tealeb S. M. Mahmoud M. Abd El-Monum Ramdan Hassan I. El-Syaed K. Saker 《Journal of Geodesy》2001,75(7-8):343-348
The use of GPS for height control in an area with existing levelling data requires the determination of a local geoid and
the bias between the local levelling datum and the one implicitly defined when computing the local geoid. If only scarse gravity
data are available, the heights of new data may be collected rapidly by determining the ellipsoidal height by GPS and not
using orthometric heights. Hence the geoid determination has to be based on gravity disturbances contingently combined with
gravity anomalies. Furthermore, existing GPS/levelling data may also be used in the geoid determination if a suitable general
gravity field modelling method (such as least-squares collocation, LSC) is applied. A comparison has been made in the Aswan
Dam area between geoids determined using fast Fourier transform (FFT) with gravity disturbances exclusively and LSC using
only the gravity disturbances and the disturbances combined with GPS/levelling data. The EGM96 spherical harmonic model was
in all cases used in a remove–restore mode. A total of 198 gravity disturbances spaced approximately 3 km apart were used,
as well as 35 GPS/levelling points in the vicinity and on the Aswan Dam. No data on the Nasser Lake were available. This gave
difficulties when using FFT, which requires the use of gridded data. When using exclusively the gravity disturbances, the
agreement between the GPS/levelling data were 0.71 ± 0.17 m for FFT and 0.63 ± 0.15 for LSC. When combining gravity disturbances
and GPS/levelling, the LSC error estimate was ±0.10 m. In the latter case two bias parameters had to be introduced to account
for a possible levelling datum difference between the levelling on the dam and that on the adjacent roads.
Received: 14 August 2000 / Accepted: 28 February 2001 相似文献
16.
The single- and dual-satellite crossover (SSC and DSC) residuals between and among Geosat, TOPEX/Poseidon (T/P), and ERS
1 or 2 have been used for various purposes, applied in geodesy for gravity field accuracy assessments and determination as
well as in oceanography. The theory is presented and various examples are given of certain combinations of SSC and DSC that test for residual altimetry data errors, mostly of non-gravitational origin, of the order of a few centimeters.
There are four types of basic DSCs and 12 independent combinations of them in pairs which have been found useful in the present
work. These are defined in terms of the `mean' and `variable' components of a satellite's geopotential orbit error from Rosborough's
1st-order analytical theory. The remaining small errors, after all altimeter data corrections are applied and the relative
offset of coordinate frames between altimetry missions removed, are statistically evaluated by means of the Student distribution.
The remaining signal of `non-gravitational' origin can in some cases be attributed to the main ocean currents which were not
accounted for among the media or sea-surface corrections. In future, they may be resolved by a long-term global circulation
model. Experience with two current models, neither of which are found either to cover the most critical missions (Geosat &
TOPEX/Poseidon) or to have the accuracy and resolution necessary to account for the strongest anomalies found across them,
is described. In other cases, the residual signal is due to errors in tides, altimeter delay corrections or El Ni?o. (Various
examples of these are also presented.) Tests of the combinations of the JGM 3-based DSC residuals show that overall the long-term
data now available are well suited for a gravity field inversion refining JGM 3 for low- and resonant-order geopotential harmonics
whose signatures are clearly seen in the basic DSC and SSC sets.
Received: 15 January 1999 / Accepted: 9 September 1999 相似文献
17.
The recovery of a full set of gravity field parameters from satellite gravity gradiometry (SGG) is a huge numerical and computational
task. In practice, parallel computing has to be applied to estimate the more than 90 000 harmonic coefficients parameterizing
the Earth's gravity field up to a maximum spherical harmonic degree of 300. Three independent solution strategies (preconditioned
conjugate gradient method, semi-analytic approach, and distributed non-approximative adjustment), which are based on different
concepts, are assessed and compared both theoretically and on the basis of a realistic-as-possible numerical simulation regarding
the accuracy of the results, as well as the computational effort. Special concern is given to the correct treatment of the
coloured noise characteristics of the gradiometer. The numerical simulations show that the three methods deliver nearly identical
results—even in the case of large data gaps in the observation time series. The newly proposed distributed non-approximative
adjustment approach, which is the only one of the three methods that solves the inverse problem in a strict sense, also turns
out to be a feasible method for practical applications.
Received: 17 December 2001 / Accepted: 17 July 2002
Acknowledgments. We would like to thank Prof. W.-D. Schuh, Institute of Theoretical Geodesy, University of Bonn, for providing us with the
serial version of the PCGMA algorithm, which forms the basis for the parallel PCGMA package developed at our institute. This
study was partially performed in the course of the GOCE project `From E?tv?s to mGal+', funded by the European Space Agency
(ESA) under contract No. 14287/00/NL/DC.
Correspondence to: R. Pail 相似文献
18.
When standard boundary element methods (BEM) are used in order to solve the linearized vector Molodensky problem we are confronted with
two problems: (1) the absence of O(|x|−2) terms in the decay condition is not taken into account, since the single-layer ansatz, which is commonly used as representation
of the disturbing potential, is of the order O(|x|−1) as x→∞. This implies that the standard theory of Galerkin BEM is not applicable since the injectivity of the integral operator
fails; (2) the N×N stiffness matrix is dense, with N typically of the order 105. Without fast algorithms, which provide suitable approximations to the stiffness matrix by a sparse one with O(N(logN)
s
), s≥0, non-zero elements, high-resolution global gravity field recovery is not feasible. Solutions to both problems are proposed.
(1) A proper variational formulation taking the decay condition into account is based on some closed subspace of co-dimension
3 of the space of square integrable functions on the boundary surface. Instead of imposing the constraints directly on the
boundary element trial space, they are incorporated into a variational formulation by penalization with a Lagrange multiplier.
The conforming discretization yields an augmented linear system of equations of dimension N+3×N+3. The penalty term guarantees the well-posedness of the problem, and gives precise information about the incompatibility
of the data. (2) Since the upper left submatrix of dimension N×N of the augmented system is the stiffness matrix of the standard BEM, the approach allows all techniques to be used to generate
sparse approximations to the stiffness matrix, such as wavelets, fast multipole methods, panel clustering etc., without any
modification. A combination of panel clustering and fast multipole method is used in order to solve the augmented linear system
of equations in O(N) operations. The method is based on an approximation of the kernel function of the integral operator by a degenerate kernel
in the far field, which is provided by a multipole expansion of the kernel function. Numerical experiments show that the fast
algorithm is superior to the standard BEM algorithm in terms of CPU time by about three orders of magnitude for N=65 538 unknowns. Similar holds for the storage requirements. About 30 iterations are necessary in order to solve the linear
system of equations using the generalized minimum residual method (GMRES). The number of iterations is almost independent
of the number of unknowns, which indicates good conditioning of the system matrix.
Received: 16 October 1999 / Accepted: 28 February 2001 相似文献
19.
C. Vigny J. Chéry T. Duquesnoy F. Jouanne J. Ammann M. Anzidei J.-P. Avouac F. Barlier R. Bayer P. Briole E. Calais F. Cotton F. Duquenne K. L. Feigl G. Ferhat M. Flouzat J.-F. Gamond A. Geiger A. Harmel M. Kasser M. Laplanche M. Le Pape J. Martinod G. Ménard B. Meyer J.-C. Ruegg J.-M. Scheubel O. Scotti G. Vidal 《Journal of Geodesy》2002,76(2):63-76
The Western Alps are among the best studied collisional belts with both detailed structural mapping and also crustal geophysical
investigations such as the ECORS and EGT seismic profile. By contrast, the present-day kinematics of the belt is still largely
unknown due to small relative motions and the insufficient accuracy of the triangulation data. As a consequence, several tectonic
problems still remain to be solved, such as the amount of N–S convergence in the Occidental Alps, the repartition of the deformation
between the Alpine tectonic units, and the relation between deformation and rotation across the Alpine arc. In order to address
these problems, the GPS ALPES group, made up of French, Swiss and Italian research organizations, has achieved the first large-scale
GPS surveys of the Western Alps. More than 60 sites were surveyed in 1993 and 1998 with a minimum observation of 3 days at
each site. GPS data processing has been done by three independent teams using different software. The different solutions
have horizontal repeatabilities (N–E) of 4–7 mm in 1993 and 2–3 mm in 1998 and compare at the 3–5-mm level in position and
2-mm/yr level in velocity. A comparison of 1993 and 1998 coordinates shows that residual velocities of the GPS marks are generally
smaller than 2 mm/yr, precluding a detailed tectonic interpretation of the differential motions. However, these data seem
to suggest that the N–S compression of the Western Alps is quite mild (less than 2 mm/yr) compared to the global convergence
between the African and Eurasian plate (6 mm/yr). This implies that the shortening must be accomodated elsewhere by the deformation
of the Maghrebids and/or by rotations of Mediterranean microplates. Also, E–W velocity components analysis supports the idea
that E–W extension exists, as already suggested by recent structural and seismotectonic data interpretation.
Received: 27 November 2000 / Accepted: 17 September 2001 相似文献
20.
An efficient algorithm is proposed for gravity field recovery from Gravity Field and Steady-State Ocean Circulation Explorer
(GOCE) satellite gravity gradient observations. The mathematical model is formulated in the time domain, which allows the
inclusion of realistic observational noise models. The algorithm combines the iterative solution of the normal equations,
using a Richardson-type iteration scheme, with the fast computation of the right-hand side of the normal equations in each
iteration step by a suitable approximation of the design matrix. The convergence of the iteration is investigated, error estimates
are provided, and the unbiasedness of the method is proved. It is also shown that the method does not converge to the solution
of the normal equations. The performance of the approach for white noise and coloured noise is demonstrated along a simulated
GOCE orbit up to spherical harmonic degree and order 180. The results also indicate that the approximation error may be neglected.
Received: 30 November 1999 / Accepted: 31 May 2000 相似文献