Various techniques and procedures for refining ERS-1 orbits

The satellite ERS-1 was launched in July 1991 in a period of high solar activity. Sparse laser tracking and the failure of the experimental microwave system (PRARE) compounded the orbital errors which resulted from mismodelling of atmospheric density and hence surface forces. Three attempts are presented here to try and refine the coarse laser orbits of ERS-1, made prior to the availability of the full altimetric dataset. The results of the first attempt indicate that by geometrically modelling the satellite shape some improvement in orbital precision may be made for any satellite; especially one where no area tables already exist. The second and third refinement attempts are based on the introduction of data from some second satellite; in these examples SPOT-2 and TOPEX/Poseidon are employed. With SPOT-2 the method makes use of the orbital similarities to produce along-track corrections for the more fully tracked SPOT-2. Transferring these corrections to ERS-1 produces improvements in the precise orbits thus determined. With TOPEX/Poseidon the greater altitude results in a more precise orbit (gravity field and atmospheric errors are of less importance). Thus, by computing height differences at crossover points of the TOPEX/Poseidon and ERS-1 ground tracks the poorer orbit of ERS-1 may be improved by the addition of derived radial corrections. In the positive light of all three results several potential modification are suggested and some further avenues of investigation indicated.

Drag coefficients with applications to satellite orbits

In the last twenty or so years the results of theory and experiment have produced much information on the characteristics of gas-surface interactions relevant to a satellite in hyperthermal free-molecular flow. This thesis contains reviews of the rarefied gas dynamics applicable to satellites and has attempted to compare existing models of gas-surface interaction with contemporary knowledge of such systems. It is shown that a more natural approach would be to characterise the gas-surface interaction using the normal and tangential momentum accommodation coefficients, igma' and igma respectively, specifically in the form igma = constant , igma' = igma'0 -igma'1sec i where i is the angle subtended between the incident flow and the surface normal and igma,igma'0 and igma'1 are constants. Adopting these relationships, the effects of atmospheric lift on inclination, i, and atmospheric drag on the semi-major axis, a, and eccentricity, e, have been investigated. Applications to ANS-1 (1974-70A) show that the observed perturbation in i can be ascribed primarily to non-zero igma'1 whilst perturbations in a and e produce constraint equations between the three parameters. The numerical results seem to imply that a good theoretical orbit is achieved despite a much lower drag coefficient than anticipated by earlier theories.

Application of ERS-1 and Topex/Poseidon altimetry using precise orbits

The work described in this thesis concerns the application of radar altimetry, collected from the ERS-1 and TOPEX/POSEIDON missions, to precise satellite orbits computed at Aston University. The data is analysed in a long arc fashion to determine range biases, time tag biases, sea surface topographies and to assess the radial accuracy of the generated orbits through crossover analysis. A sea surface variability study is carried out for the North Sea using repeat altimeter profiles from ERS-1 and TOPEX/POSEIDON in order to verify two local U.K. models for ocean tide and storm surge effects. An on-side technique over the English Channel is performed to compute the ERS-1, TOPEX and POSEIDON altimeter range biases by using a combination of altimetry, precise orbits determined by short arc methods, tide gauge data, GPS measurements, geoid, ocean tide and storm surge models. The remaining part of the thesis presents some techniques for the short arc correction of long arc orbits. Validation of this model is achieved by way of comparison with actual SEASAT short arcs. Simulations are performed for the ERS-1 microwave tracking system, PRARE, using the range data to determine time dependent orbit corrections. Finally, a brief chapter is devoted to the recovery of errors in station coordinates by the use of multiple short arcs.

A variational method and approximations of a Cauchy problem for elliptic equations

A Cauchy problem for general elliptic second-order linear partial differential equations in which the Dirichlet data in H½(?1 ? ?3) is assumed available on a larger part of the boundary ? of the bounded domain O than the boundary portion ?1 on which the Neumann data is prescribed, is investigated using a conjugate gradient method. We obtain an approximation to the solution of the Cauchy problem by minimizing a certain discrete functional and interpolating using the finite diference or boundary element method. The minimization involves solving equations obtained by discretising mixed boundary value problems for the same operator and its adjoint. It is proved that the solution of the discretised optimization problem converges to the continuous one, as the mesh size tends to zero. Numerical results are presented and discussed.

An iterative method based on boundary integrals for elliptic Cauchy problems in semi-infinite domains

In this study, we investigate the problem of reconstruction of a stationary temperature field from given temperature and heat flux on a part of the boundary of a semi-infinite region containing an inclusion. This situation can be modelled as a Cauchy problem for the Laplace operator and it is an ill-posed problem in the sense of Hadamard. We propose and investigate a Landweber-Fridman type iterative method, which preserve the (stationary) heat operator, for the stable reconstruction of the temperature field on the boundary of the inclusion. In each iteration step, mixed boundary value problems for the Laplace operator are solved in the semi-infinite region. Well-posedness of these problems is investigated and convergence of the procedures is discussed. For the numerical implementation of these mixed problems an efficient boundary integral method is proposed which is based on the indirect variant of the boundary integral approach. Using this approach the mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing that stable and accurate reconstructions of the temperature field on the boundary of the inclusion can be obtained also in the case of noisy data. These results are compared with those obtained with the alternating iterative method.

An iterative procedure for solving a Cauchy problem for second order elliptic equations

An iterative method for reconstruction of solutions to second order elliptic equations by Cauchy data given on a part of the boundary, is presented. At each iteration step, a series of mixed well-posed boundary value problems are solved for the elliptic operator and its adjoint. The convergence proof of this method in a weighted L2 space is included. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical solution of an elliptic 3-dimensional Cauchy problem by the alternating method and boundary integral equations

We consider the Cauchy problem for the Laplace equation in 3-dimensional doubly-connected domains, that is the reconstruction of a harmonic function from knowledge of the function values and normal derivative on the outer of two closed boundary surfaces. We employ the alternating iterative method, which is a regularizing procedure for the stable determination of the solution. In each iteration step, mixed boundary value problems are solved. The solution to each mixed problem is represented as a sum of two single-layer potentials giving two unknown densities (one for each of the two boundary surfaces) to determine; matching the given boundary data gives a system of boundary integral equations to be solved for the densities. For the discretisation, Weinert's method [24] is employed, which generates a Galerkin-type procedure for the numerical solution via rewriting the boundary integrals over the unit sphere and expanding the densities in terms of spherical harmonics. Numerical results are included as well.