Stochastic perturbation analysis of groundwater flow. Spatially variable soils,semi-infinite domains and large fluctuations

As is well known, a complete stochastic solution of the stochastic differential equation governing saturated groundwater flow leads to an infinite hierarchy of equations in terms of higher-order moments. Perturbation techniques are commonly used to close this hierarchy, using power-series expansions. These methods are applied by truncating the series after a finite number of terms, and products of random gradients of conductivity and head potential are neglected. Uncertainty regarding the number or terms required to yield a sufficiently accurate result is a significant drawback with the application of power series-based perturbation methods for such problems. Low-order series truncation may be incapable of representing fundamental characteristics of flow and can lead to physically unreasonable and inaccurate solutions of the stochastic flow equation. To support this argument, one-dimensional, steady-state, saturated groundwater flow is examined, for the case of a spatially distributed hydraulic conductivity field. An ordinary power-series perturbation method is used to approximate the mean head, using second-order statistics to characterize the conductivity field. Then an interactive perturbation approach is introduced, which yields improved results compared to low-order, power-series perturbation methods for situations where strong interactions exist between terms in such approximations. The interactive perturbation concept is further developed using Feynman-type diagrams and graph theory, which reduce the original stochastic flow problem to a closed set of equations for the mean and the covariance functions. Both theoretical and practical advantages of diagrammatic solutions are discussed; these include the study of bounded domains and large fluctuations.     

反演二维弹性波动方程组多参数的渐近展开法

采用摄动法对二维弹性波动方程组的系数反问题进行渐近近似,给出了反演系数的具体计算方法。用摄动法解微分方程组的系数反问题还是首次。此外,还采用了地球物理分层模型,这种分层模型便于更精细地反映地球不同圈层的物理特征,更接近于实际情况。同时对进一步提高反演分辨率、增加反演参数、对病态方程组如何克服不适定性等问题,作了具体阐述,具有重要的理论意义与应用价值。     

Groebner-basis solution of the three-dimensional resection problem (P4P)

The three-dimensional (3-D) resection problem is usually solved by first obtaining the distances connecting the unknown point P{X,Y,Z} to the known points P_i{X_i,Y_i,Z_i}i=1,2,3 through the solution of the three nonlinear Grunert equations and then using the obtained distances to determine the position {X,Y,Z} and the 3-D orientation parameters {,, }. Starting from the work of the German J. A. Grunert (1841), the Grunert equations have been solved in several substitutional steps and the desire as evidenced by several publications has been to reduce these number of steps. Similarly, the 3-D ranging step for position determination which follows the distance determination step involves the solution of three nonlinear ranging (`Bogenschnitt') equations solved in several substitution steps. It is illustrated how the algebraic technique of Groebner basis solves explicitly the nonlinear Grunert distance equations and the nonlinear 3-D ranging (`Bogenschnitt') equations in a single step once the equations have been converted into algebraic (polynomial) form. In particular, the algebraic tool of the Groebner basis provides symbolic solutions to the problem of 3-D resection. The various forward and backward substitution steps inherent in the classical closed-form solutions of the problem are avoided. Similar to the Gauss elimination technique in linear systems of equations, the Groebner basis eliminates several variables in a multivariate system of nonlinear equations in such a manner that the end product normally consists of a univariate polynomial whose roots can be determined by existing programs e.g. by using the roots command in Matlab.Acknowledgments.The first author wishes to acknowledge the support of JSPS (Japan Society of Promotion of Science) for the financial support that enabled the completion of the write-up of the paper at Kyoto University, Japan. The author is further grateful for the warm welcome and the good working atmosphere provided by his hosts Professors S. Takemoto and Y. Fukuda of the Department of Geophysics, Graduate School of Science, Kyoto University, Japan.     

Multiple Parameter Regularization: Numerical Solutions and Applications to the Determination of Geopotential from Precise Satellite Orbits

Kaula’s rule of thumb has been used in producing geopotential models from space geodetic measurements, including the most recent models from satellite gravity missions CHAMP. Although Xu and Rummel (Manuscr Geod 20 8–20, 1994b) suggested an alternative regularization method by introducing a number of regularization parameters, no numerical tests have ever been conducted. We have compared four methods of regularization for the determination of geopotential from precise orbits of COSMIC satellites through simulations, which include Kaula’s rule of thumb, one parameter regularization and its iterative version, and multiple parameter regularization. The simulation results show that the four methods can indeed produce good gravitational models from the precise orbits of centimetre level. The three regularization methods perform much better than Kaula’s rule of thumb by a factor of 6.4 on average beyond spherical harmonic degree 5 and by a factor of 10.2 for the spherical harmonic degrees from 8 to 14 in terms of degree variations of root mean squared errors. The maximum componentwise improvement in the root mean squared error can be up to a factor of 60. The simplest version of regularization by multiplying a positive scalar with a unit matrix is sufficient to better determine the geopotential model. Although multiple parameter regularization is theoretically attractive and can indeed eliminate unnecessary regularization for some of the harmonic coefficients, we found that it only improved its one parameter version marginally in this COSMIC example in terms of the mean squared error.     

顾及地球引力摄动的卫星和卫星星下点位置计算

本文叙述了顾及地球引力摄动的卫星化置计算方法；利用天球坐标系与地球坐标系之间的坐标变换关系，给出了由卫星位置计算卫星星下点位置的严密计算公式。     

Ground water contaminant transport by nondivergence-free, unsteady and nonstationary velocity fields

Pore flow velocity is assumed to be a nondivergence-free, unsteady, and nonstationary random function of space and time for ground water contaminant transport in a heterogeneous medium. The laboratory-scale stochastic contaminant transport equation is up scaled to field scale by taking the ensemble average of the equation by using the cumulant expansion method. A new velocity correction, which is a function of mean pore flow velocity divergence, is obtained due to strict second order cumulant expansion (without omitting any term after the expansion). The field scale transport equations under the divergence-free pore flow velocity field assumption are also derived by simplifying the nondivergence-free field scale equation. The significance of the new velocity correction term is investigated on a two dimensional transport problem driven by a density dependent flow.     

Errors due to the anisotropic-common-ray approximation of the coupling ray theory

The common-ray approximation eliminates problems with ray tracing through S-wave singularities and also considerably simplifies the numerical algorithm of the coupling ray theory for S waves, but may introduce errors in travel times due to the perturbation from the common reference ray. These travel-time errors can deteriorate the coupling-ray-theory solution at high frequencies. It is thus of principal importance for numerical applications to estimate the errors due to the common-ray approximation applied. The anisotropic-common-ray approximation of the coupling ray theory is more accurate than the isotropic-common-ray approximation. We derive the equations for estimating the travel-time errors due to the anisotropic-common-ray (and also isotropic-common-ray) approximation of the coupling ray theory. The errors of the common-ray approximations are calculated along the anisotropic common rays in smooth velocity models without interfaces. The derivation is based on the general equations for the second-order perturbations of travel time.     

Progress in Predictability Studies in China （2003-2006）

Since the last International Union of Geodesy and Geophysics General Assembly(2003),predictability studies in China have made significant progress.For dynamic forecasts,two novel approaches of conditional nonlinear optimal perturbation and nonlinear local Lyapunov exponents were proposed to cope with the predictability problems of weather and climate,which are superior to the corresponding linear theory.A possible mechanism for the"spring predictability barrier"phenomenon for the El Ni(?)o-Southern Oscillation (ENSO)was provided based on a theoretical model.To improve the forecast skill of an intermediate coupled ENSO model,a new initialization scheme was developed,and its applicability was illustrated by hindcast experiments.Using the reconstruction phase space theory and the spatio-temporal series predictive method, Chinese scientists also proposed a new approach to improve dynamical extended range(monthly)prediction and successfully applied it to the monthly-scale predictability of short-term climate variations.In statistical forecasts,it was found that the effects of sea surface temperature on precipitation in China have obvious spatial and temporal distribution features,and that summer precipitation patterns over east China are closely related to the northern atmospheric circulation.For ensemble forecasts,a new initial perturbation method was used to forecast heavy rain in Guangdong and Fujian Provinces on 8 June 1998.Additionally, the ensemble forecast approach was also used for the prediction of a tropical typhoons.A new downscaling model consisting of dynamical and statistical methods was provided to improve the prediction of the monthly mean precipitation.This new downsealing model showed a relatively higher score than the issued operational forecast.     

Hybridization of the probability perturbation method with gradient information

Geostatistically based history-matching methods make it possible to devise history-matching strategies that will honor geologic knowledge about the reservoir. However, the performance of these methods is known to be impeded by slow convergence rates resulting from the stochastic nature of the algorithm. It is the purpose of this paper to introduce a method that integrates qualitative gradient information into the probability perturbation method to improve convergence. The potential of the proposed method is demonstrated on a synthetic history-matching example. The results indicate that inclusion of qualitative gradient information improves the performance of the probability perturbation method.