Algebraic Solution of GPS Pseudo-Ranging Equations     

Joseph L. Awange  Erik W. Grafarend 《GPS Solutions》2002,5(4):20-32

Several procedures for solving, in a closed form the GPS pseudo-ranging four-point problem P4P in matrix form already exist. We present here alternative algebraic procedures using Multipolynomial resultant and Groebner basis to solve the same problem. The advantage is that these algebraic algorithms have already been implemented in algebraic software such as “Mathematica” and “Maple”. The procedures are straightforward and simple to apply. We illustrate here how the algebraic techniques of Multipolynomial resultant and Groebner basis explicitly solve the nonlinear GPS pseudo-ranging four-point equations once they have been converted into algebraic (polynomial) form and reduced to linear equations. In particular, the algebraic tools of Multipolynomial resultant and Groebner basis provide symbolic solutions to the GPS four-point pseudo-ranging problem. The various forward and backward substitution steps inherent in the clasical closed form solutions of the problem are avoided. Similar to the Gauss elimination techniques in linear systems of equations, the Multipolynomial resultant and Groebner basis approaches eliminate several variables in a multivariate system of nonlinear equations in such a manner that the end product normally consists of univariate polynomial equations (in this case quadratic equations for the range bias expressed algebraically using the given quantities) whose roots can be determined by existing programs (e. g., the roots command in MATLAB). © 2002 Wiley Periodicals, Inc.  相似文献   

GPS Solutions: Closed Forms, Critical and Special Configurations of P4P   总被引：1，自引：0，他引：1  

Erik W. Grafarend  Jeffrey Shan 《GPS Solutions》2002,5(3):29-41

P4P is the pseudo-ranging 4-point problem as it appears as the basic configuration of satellite positioning with pseudo-ranges as observables. In order to determine the ground receiver/satellite receiver (LEO networks) position from four positions of satellite transmitters given, a system of four nonlinear (algebraic) equations has to be solved. The solution point is the intersection of four spherical cones if the ground receiver/satellite receiver clock bias is implemented as an unknown. Here we determine the critical configuration manifold (Determinantal Loci, Inverse Function Theorem, Jacobi map) where no solution of P4P exists. Four examples demonstrate the critical linear manifold. The algorithm GS solves in a closed form P4P in a manner similar to Groebner bases: The algebraic nonlinear observational equations are reduced in the forward step to one quadratic equation in the clock bias unknown. In the backward step two solutions of the position unknowns are generated in closed form. Prior information in P4P has to be implemented in order to decide which solution is acceptable. Finally, the main target of our contribution is formulated: Can we identify a special configuration of satellite transmitters and ground receiver/satellite receiver where the two solutions are reduced to one. A special geometric analysis of the discriminant solves this problem. ? 2002 Wiley Periodicals, Inc.  相似文献   

Functional assessment of a reference wetland set as a tool for science, management and restoration   总被引：1，自引：0，他引：1  

Stuart E. G. Findlay  Erik Kiviat  W. Charles Nieder  Elizabeth A. Blair 《Aquatic Sciences - Research Across Boundaries》2002,64(2):107-117

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Elektronenmikroskopische Untersuchungen an mikritischen Kalken     

Prof. Dr. Erik Flügel 《International Journal of Earth Sciences》1967,56(1):341-358

Zusammenfassung Es werden die Entstehungsmöglichkeiten von mikritischen Kalken diskutiert und an Hand von elektronenmikroskopischen Aufnahmen Hinweise auf genetische Interpretationen vorgeführt.

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The origin of micritic limestones is discussed and genetic interpretation of micrites from different environments, studied by electron microscope, is given.

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Résumé L'origine des calcaires micritic est discutée et une interprétation génétique des micrites de milieux différents est donnée à l'aide de la microscopie électronique.

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Vorgetragen auf der 56. Jahresversammlung der Geologischen Vereinigung in Wien im Februar 1966.  相似文献   

Reconstructing the historical changes in Daphnia mean size and planktivorous fish abundance in lakes from the size of Daphnia ephippia in the sediment   总被引：3，自引：0，他引：3  

Erik Jeppesen  Jens Peder Jensen  Susanne Amsinck  Frank Landkildehus  Torben Lauridsen  Stuart F. Mitchell 《Journal of Paleolimnology》2002,27(1):133-143

The zooplankton community structure in lakes is highly influenced by size-selective predation by fish, with small zooplankton species dominating at high predation pressure. Remains of cladocerans are preserved in the sediment and may be used to trace historical changes in fish predation. We determined how contemporary data on planktivorous fish were related to the size of Daphnia ephippia (dorsal length) in the surface sediment (0-1 cm) of 52 mainly shallow lakes with contrasting densities of fish and nutrients (TP: 0.002-0.60 mg P l^-1). Density of fish expressed as catch per unit effort, in terms of numbers in multiple mesh-sized gill nets (CPUE_n), decreased significantly with increasing mean size of ephippia. The relationship was improved by adding TP as an independent variable, now explaining 90% of the variation in CPUE_n on the full data set covering lakes in Denmark, Greenland and New Zealand, and 78% if only data on Danish lakes were used. CPUE by weight of planktivorous fish and mean weight of Daphnia in the pelagial during summer were also related to ephippial size. By including contemporary data on established relationships between the sizes of egg-bearing female Daphnia and ephippia, we inferred changes in the CPUE_n, mean size of ephippia-bearing Daphnia and summer mean body weight of Daphnia from ephippial size in four lakes during the past 1-2 centuries. In a hypertrophic lake subject to periodic fish kills, Daphnia mean body weight was high and CPUE_n was low compared with those in two eutrophic lakes, while CPUE_n was low and Daphnia body weight was high in the least eutrophic, clearwater lake. Estimated CPUE_n and Daphnia mean weight in the surface sediment of these four lakes corresponded well with contemporary data. Only small changes in ephippial size with time were observed in the clearwater lake and in one of the lakes that had suffered early eutrophication, while major changes occurred in the two other lakes that had been subjected to a major increase in nutrient input or fish kills. We conclude that Daphnia ephippia preserved in the surface sediments of lakes may be a useful and efficient method to quantify the present-day abundance of planktivorous fish and Daphnia mean size. The method is particularly valid in surveys aimed to give a general picture of the fish stock and the ecological state in a set of lakes in a region rather than a precise estimate for a single lake. Though some evidence is provided, more work is needed to evaluate whether the equations are valid for hind-casting in down-core palaeoecological studies.  相似文献   

Satellite and surface temperature data at odds?     

James Hansen  Helene Wilson  Makiko Sato  Reto Ruedy  Kathy Shah  Erik Hansen 《Climatic change》1995,30(1):103-117

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On lateral viscosity contrast in the mantle and the rheology of low-frequency geodynamics   总被引：2，自引：0，他引：2  

Erik R. Ivins  Charles G. Sammis 《Geophysical Journal International》1995,123(2):305-322

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Statistics and geometry of the eigenspectra of three-dimensional second-rank symmetric random tensors   总被引：1，自引：0，他引：1  

Peiliang Xu  Erik Grafarend 《Geophysical Journal International》1996,127(3):744-756

Geoscientists have undertaken mapping of the Earth's crustal strain (or stress) fields using a great variety of field data. The output can be represented by a 3-D second-rank symmetric random strain tensor. The random principal strains-land rotations of the random tensor are frequently computed. The accuracy is calculated using a first-order approximation. The distribution aspects of the random principal strains and rotations have received almost no attention in Earth Sciences. A first-order approximation of accuracy may not be sufficient if the signal-to-noise ratio is small, as is often the case for geodetically derived random strain tensors. Therefore, the purpose of this paper is to investigate the distribution and estimation problems of the general 3-D second-rank tensor equation GΛG ^T= T , where T is a given 3-D second-rank symmetric random tensor, Λ a diagonal (3 × 3) random eigenvalue matrix, and G a (3 × 3) random orientation matrix, which is also orthogonal. Λ and G are to be estimated (or solved) from T . If some eigenvalues coincide, additional conditions are imposed on the eigenvectors so that they can be chosen uniquely. The joint probability density function (pdf) of the random eigenvalues and rotations will be worked out, given a joint pdf of the elements of random tensors T. Because the rotations are of special interest in Earth Sciences, we shall also derive the joint marginal pdf of random rotations. The geometry of eigenspectra will be studied. The biases of random eigenvalues and rotations will be derived, which have been neglected in the past. They can be very crucial in interpreting the pattern of a derived strain field, however, when applied to a real Earth Science problem. The variance-covariance matrices will be computed using a second-order approximation.  相似文献   

WEGS—A working club of Western European geological surveys     

Erik Stenestad 《Engineering Geology》1990,29(4):393-397

WEGS was established in 1971 as a working club of all the directors of the geological surveys in Western Europe. The purpose is to encourage the exchange of information between the 21 member-surveys representing all of the Western European countries.

WEGS has annual plenary meetings convened by all member countries in turn. The general exchange of information includes the surveys' relationships to national and international organisations, institutions and large scale projects. Also on the agenda are reports from WEGS Working Groups on Environmental Geology, Geochemistry and Computers. At each meeting the opportunity is taken during field trips to examine the local geological conditions and problems related to geology.

Current activities of WEGS include initiatives concerning standardisation and transfer of data between the member-surveys, a guideline book on environmental geology with case histories from all parts of Western Europe and possible joint activities within marine geology.  相似文献   

An approximate method for calculating the depth of a surface-reaching gravitational mass-anomaly,giving a Bouguer-field with radial or axial symmetry     

Tengström  Erik 《Pure and Applied Geophysics》1960,46(1):23-25

Summary The author describes a method of determining the size and shape of a gravitating disturbance, treated as a surface-reaching a) vertical cylinder, b) infinite parallelepipedon, when the local Bouguer anomaly field has a) radial symmetry, b) axial symmetry. He introduces the quantity which e. g. serves the purpose of distinguishing very easily between deepseated and shallow disturbances.Presented at the IUGG-Meeting, Helsinki 1960 (IAG, Sec. V).  相似文献