On the solution of the Molodenskii boundary value problem using successive approximations

Gravimetric methods of determining the figure of the Earth can be divided into two groups. The first employs an auxiliary surface—the geoid. The second employs the so-called quasigeoid as an auxiliary surface. Its main advantage is in that it only uses surface measurements and that it does not require knowledge of the structure of the Earth's crust. This method was treated in [1]. The results display an accuracy of the order of the Earth's flattening. The purpose of this paper is to show that this method can be applied theoretically to reach an arbitrary accuracy.     

卫星测高混合边值问题的球谐级数解法

研究了球界面下卫星测高问题的解法,利用有限逼近方法得到了下列结论：若陆地部分是球冠,则卫星测高问题的解可以转换成关于球谐级数位系数的线性方程组,并且位系数的阶和次是以分离形式出现的,从而确保该解法具有实用意义. 利用重力场360阶模型进行模拟计算的结果表明：该解法得到的位系数的相对精度达到了10－１１. 同时证明了常用的Stokes问题、Dirichlet问题、Neumann问题可以看成卫星测高问题的特殊情况.     

A Brovar-type solution of the fixed geodetic boundary-value problem

For more than 150 years gravity anomalies have been used for the determination of geoidal heights, height anomalies and the external gravity field. Due to the fact that precise ellipsoidal heights could not be observed directly, traditionally a free geodetic boundary-value problem (GBVP) had to be formulated which after linearisation is related to gravity anomalies. Since nowadays the three-dimensional positions of gravity points can be determined by global navigation satellite systems very precisely, the modern formulation of the GBVP can be based on gravity disturbances which are related to a fixed GBVP using the known topographical surface of the Earth as boundary surface. The paper discusses various approaches into the solution of the fixed GBVP which after linearization corresponds to an oblique-derivative boundary-value problem for the Laplace equation. Among the analytical solution approaches a Brovar-type solution is worked out in detail, showing many similarities with respect to the classical solution of the scalar free GBVP.     

Numerical solution of the two-dimensional inverse geomagnetic induction problem

Summary An effective numerical approach to the solution of the two-dimensional inverse geomagnetic induction problem using the linearization method is presented. The numerical realization of the inversion is based on Marquardt's algorithm, for which the solution of the direct problem and the partial derivatives of this solution with respect to the electrical parameters of the medium are computed by the finite difference method. Theoretical models are studied and numerical results are presented.     

On the solution of the boundary problem of the potential theory by means of the vertical derivative of the gravity anomaly

Summary The boundary problem of the potential theory is treated for the case in which the values of the vertical derivative of the gravity anomaly are known. There boundary conditions are derived and alternatives are also treated.Dedicated to RNDr. Jan Pícha, CSc., on his 60th Birthday     

Spherical tensor approach to the solution of the mantle stress problem

Summary The technique of spherical tensors is applied to the problem of stress in the mantle. An efficient method for stress computation is developed for the Newtonian mantle with spherical boundaries assuming that the density and gravitational potential in the form of harmonic expansions are known. The ways of including more complicated rheological models as well as the non-spherical shape of boundaries are outlined.

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The analytic solution of the Shallow-Water Equations with partially open sluice-gates: The dam-break problem

The general solution of the dam-break problem with partial uplift of the sluice-gate is presented in the framework of the one-dimensional Shallow-Water Equations, under the hypothesis that the classic Energy–Momentum (E–M) formulation is used to evaluate the flow characteristics at the gate. Due to the nonlinearity of the problem, there are ranges of the initial conditions for which the problem admits multiple solutions, or no solution. In the ranges in which there are multiple solutions, the maximization of the discharge under the gate is used as a disambiguation criterion. The exact solutions are used as a benchmark in order to evaluate the results of a simple Finite-Volume scheme, where the discharge under the gate and the forces exerted on the flow by the gate itself are calculated using the E–M formulation.     

On the structure of normal spiral galaxies with finite disk thickness part III: The asymptotic solution and the quantum condition of the general eigenvalue problem

Abstract

The asymptotic solution and the quantum condition of an eigen-value problem which appears in the density wave theory of the spiral galaxy are re-examined from a mathematical point of view. The dependence of the complex eigen-frequency ω on a large parameter λ is carefully discussed. It turns out that, in the expansion ω=∑^χ _i=o ωj/λj, the first term ωo must be real in order that the radiation condition may be satisfied. A related conclusion is that the quantum condition may be expressed in terms of an integral along the real axis of the complex plane of the independent variable. This form for the complex quantum condition is convenient not only for determining the complex eigen-frequency numerically, but also for drawing some physically interesting conclusions directly.     

The stability problem of reverse time migration for viscoacoustic VTI media

In real strata anisotropy and viscosity extensively exists. They degraded waveforms in amplitude, resulting in which reducing of image resolution. To obtain high-precision imaging of deep reservoirs, we extended the separated viscous and anisotropic reverse time migration (RTM) to a stable viscoacoustic anisotropic RTM for vertical transverse isotropic (VTI) media, based on single generalized standard and linear solid (GSLS) media theory.. We used a pseudo-spectral method to develop the numerical simulation. By introducing a regularization operator to eliminate the high-frequency instability problem, we built a stable inverse propagator and achieved viscoacoustic VTI media RTM. High-resolution imaging results were obtained after correcting for the effects of anisotropy and viscosity. Synthetic tests verify the validity and accuracy of algorithm.     

A BME solution of the inverse problem for saturated groundwater flow

In most real-world hydrogeologic situations, natural heterogeneity and measurement errors introduce major sources of uncertainty in the solution of the inverse problem. The Bayesian Maximum Entropy (BME) method of modern geostatistics offers an efficient solution to the inverse problem by first assimilating various physical knowledge bases (hydrologic laws, water table elevation data, uncertain hydraulic resistivity measurements, etc.) and then producing robust estimates of the subsurface variables across space. We present specific methods for implementing the BME conceptual framework to solve an inverse problem involving Darcys law for subsurface flow. We illustrate one of these methods in the case of a synthetic one-dimensional case study concerned with the estimation of hydraulic resistivity conditioned on soft data and hydraulic head measurements. The BME framework processes the physical knowledge contained in Darcys law and generates accurate estimates of hydraulic resistivity across space. The optimal distribution of hard and soft data needed to minimize the associated estimation error at a specified sampling cost is determined. This work was supported by grants from the National Institute of Environmental Health Sciences (Grant no. 5 P42 ES05948 and P30ES10126), the National Aeronautics and Space Administration (Grant no. 60-00RFQ041), the Army Research Office (Grant no. DAAG55-98-1-0289), and the National Science Foundation under Agreement No. DMS-0112069.     

Exact solution of the source problem for a finite threedimensional source

Summary In this note the surface displacement for an elastic half-space due to a finite cylindrical source has been calculated in exact terms. The source consists of outward body forces acting on the surface of the right circular cylinder, one end of which is parallel to the free surface of the half-space.     

Numerical solution of the dam failure problem by the random choice method

A numerical solution of the dam failure problem as described by the one-dimensional shallow water equations is presented. The construction of the solution is based on the random choice method consisting in the superposition of locally theoretical solutions and sampling techniques. The search of the optimal sampling is performed through the application of the random choice method to the scalar wave equation. The dam failure problem is then solved and a comparison with the theoretical solution is presented. It is shown that the random choice method computes the bore with almost infinite resolution, represents exactly the constant state behind it and calculates the depression wave with great accuracy.     

On the problem of frequency calibration of induction variometers with large air-core loops

Zusammenfassung Es wird eine Methode für experimentelle Untersuchung der Frequenzkennlinien von Induktionsvariometern mit Aufnahmeschleifen nichttraditioneller Ausmasse beschrieben. Die Grundlage des beschriebenen Verfahrens bildet eine genaue Bestimmung jenes Teils des magnetischen Kraftflusses, der durch die Aufnahmeschleife durchgeht.     

On the problem of standardization of the magnitude-frequency relation

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On one formulation of the frontal wave problem

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On the geostatistical approach to the inverse problem

The geostatistical approach to the inverse problem is discussed with emphasis on the importance of structural analysis. Although the geostatistical approach is occasionally misconstrued as mere cokriging, in fact it consists of two steps: estimation of statistical parameters (“structural analysis”) followed by estimation of the distributed parameter conditional on the observations (“cokriging” or “weighted least squares”). It is argued that in inverse problems, which are algebraically undetermined, the challenge is not so much to reproduce the data as to select an algorithm with the prospect of giving good estimates where there are no observations. The essence of the geostatistical approach is that instead of adjusting a grid-dependent and potentially large number of block conductivities (or other distributed parameters), a small number of structural parameters are fitted to the data. Once this fitting is accomplished, the estimation of block conductivities ensues in a predetermined fashion without fitting of additional parameters. Also, the methodology is compared with a straightforward maximum a posteriori probability estimation method. It is shown that the fundamental differences between the two approaches are: (a) they use different principles to separate the estimation of covariance parameters from the estimation of the spatial variable; (b) the method for covariance parameter estimation in the geostatistical approach produces statistically unbiased estimates of the parameters that are not strongly dependent on the discretization, while the other method is biased and its bias becomes worse by refining the discretization into zones with different conductivity.