The development of stochastic space transformation and diagrammatic perturbation techniques in subsurface hydrology

This paper develops concepts and methods to study stochastic hydrologic models. Problems regarding the application of the existing stochastic approaches in the study of groundwater flow are acknowledged, and an attempt is made to develop efficient means for their solution. These problems include: the spatial multi-dimensionality of the differential equation models governing transport-type phenomena; physically unrealistic assumptions and approximations and the inadequacy of the ordinary perturbation techniques. Multi-dimensionality creates serious mathematical and technical difficulties in the stochastic analysis of groundwater flow, due to the need for large mesh sizes and the poorly conditioned matrices arising from numerical approximations. An alternative to the purely computational approach is to simplify the complex partial differential equations analytically. This can be achieved efficiently by means of a space transformation approach, which transforms the original multi-dimensional problem to a much simpler unidimensional space. The space transformation method is applied to stochastic partial differential equations whose coefficients are random functions of space and/or time. Such equations constitute an integral part of groundwater flow and solute transport. Ordinary perturbation methods for studying stochastic flow equations are in many cases physically inadequate and may lead to questionable approximations of the actual flow. To address these problems, a perturbation analysis based on Feynman-diagram expansions is proposed in this paper. This approach incorporates important information on spatial variability and fulfills essential physical requirements, both important advantages over ordinary hydrologic perturbation techniques. Moreover, the diagram-expansion approach reduces the original stochastic flow problem to a closed set of equations for the mean and the covariance function.     

Simple stochastic modelling of the eruption history of a basaltic volcano: Nyamuragira,Zaire

Three simple models of the behaviour of a series of basaltic eruptions have been tested against the eruptive history of Nyamuragira. The data set contains the repose periods and the volumes of lava emitted in 22 eruptions since 1901. Model 1 is fully stochastic and eruptions of any volume with random repose intervals are possible. Models 2 and 3 are constrained by deterministic limits on the maximum capacity of the magma reservoir and on the lowest drainage level of the reservoir respectively. The method of testing these models involves (1) seeking change points in the time series to determine regimes of uniform magma supply rate, and (2) applying linear regression to these regimes, which for models 2 and 3 are the determinsstic limits to those models. Two change points in the time series for Nyamuragira, in 1958 and 1980, were determined using a Kolmogorov-Smirnov technique. The latter change involved an increase in the magma supply rate by a factor of 2.5, from 0.55 to 1.37 m³s^-1. Model 2 provides the best fit to the behavior of Nyamuragira with the ratio of variation explained by the model to total variation. R², being greater than 0.9 for all three regimes. This fit can be interpreted to mean that there is a determinstic limit to the elastic strength of the magma reservoir 4–8 km below the summit of the volcano.     

Stochastic space transforms in subsurface hydrology — Part 2: Generalized spectral decompositions and plancherel representations

In earlier publications, certain applications of space transformation operators in subsurface hydrology were considered. These operators reduce the original multi-dimensional problem to the one-dimensional space, and can be used to study stochastic partial differential equations governing groundwater flow and solute transport processes. In the present work we discuss developments in the theoretical formulation of flow models with space-dependent coefficients in terms of space transformations. The formulation is based on stochastic Radon operator representations of generalized functions. A generalized spectral decomposition of the flow parameters is introduced, which leads to analytically tractable expressions of the space transformed flow equation. A Plancherel representation of the space transformation product of the head potential and the log-conductivity is also obtained. A test problem is first considered in detail and the solutions obtained by means of the proposed approach are compared with the exact solutions obtained by standard partial differential equation methods. Then, solutions of three-dimensional groundwater flow are derived starting from solutions of a one-dimensional model along various directions in space. A step-by-step numerical formulation of the approach to the flow problem is also discussed, which is useful for practical applications. Finally, the space transformation solutions are compared with local solutions obtained by means of series expansions of the log-conductivity gradient.     

Carbon and water cycling in a Bornean tropical rainforest under current and future climate scenarios

We examined how the projected increase in atmospheric CO₂ and concomitant shifts in air temperature and precipitation affect water and carbon fluxes in an Asian tropical rainforest, using a combination of field measurements, simplified hydrological and carbon models, and Global Climate Model (GCM) projections. The model links the canopy photosynthetic flux with transpiration via a bulk canopy conductance and semi-empirical models of intercellular CO₂ concentration, with the transpiration rate determined from a hydrologic balance model. The primary forcing to the hydrologic model are current and projected rainfall statistics. A main novelty in this analysis is that the effect of increased air temperature on vapor pressure deficit (D) and the effects of shifts in precipitation statistics on net radiation are explicitly considered. The model is validated against field measurements conducted in a tropical rainforest in Sarawak, Malaysia under current climate conditions. On the basis of this model and projected shifts in climatic statistics by GCM, we compute the probability distribution of soil moisture and other hydrologic fluxes. Regardless of projected and computed shifts in soil moisture, radiation and mean air temperature, transpiration was not appreciably altered. Despite increases in atmospheric CO₂ concentration (C_a) and unchanged transpiration, canopy photosynthesis does not significantly increase if C_i/C_a is assumed constant independent of D (where C_i is the bulk canopy intercellular CO₂ concentration). However, photosynthesis increased by a factor of 1.5 if C_i/C_a decreased linearly with D as derived from Leuning stomatal conductance formulation [R. Leuning. Plant Cell Environ 1995;18:339–55]. How elevated atmospheric CO₂ alters the relationship between C_i/C_a and D needs to be further investigated under elevated atmospheric CO₂ given its consequence on photosynthesis (and concomitant carbon sink) projections.     

三危山断层近场地震动对敦煌莫高窟的影响研究

三危山断层距敦煌莫高窟仅约3km。本建立横向均匀成层的地震断层模型，利用随机有限断层方法，预测三危山断层发生5．5级、6．0级、6．5级地震时地震动对莫高窟的影响。研究表明逆断层使三危山断层辐射的地震动分布形态呈现出明显的上下盘效应。同时，将计算PGA值与已有的根据其它地区地震记录推出的PGA衰减关系进行了对比分析，认为该结果可供莫高窟抗震防护加固设计时参考。     

A procedure for testing the assumption of homoscedasticity in least squares residuals: a case study of GPS carrier-phase observations

Due to increased demands on the quality of the results of Global Positioning System (GPS) evaluations, various authors have studied improvements of the stochastic model of GPS carrier-phase observations. These improvements are based on the reasonable assumption that the commonly used stochastic model with independent and homoscedastic (i.e. equal variance) errors is unrealistic. However, this has not been proved rigorously so far. A statistical test procedure based on uncorrelated least–squares residuals, which allows verification of the hypothesis of a heterogeneous variance, is provided. The statistical test procedure is of interest in its own right, and is independent of the practical problem considered. The presented technique is applied to GPS carrier-phase observations. Results show that the variances of the investigated observations are far from homogeneous. It is indicated that the error variances of the presented data increase with decreasing GPS satellite elevation. These results confirm the assumption that the commonly used stochastic model of GPS observations is inadequate and has to be improved.     

铅芯叠层橡胶支座基础隔震结构双向地震反应分析

本文对铅芯叠层橡胶支座双向耦合恢复力模型进行了改进，采用基础隔震结构动力分析程序DABIS对铅芯叠层橡胶支座基础隔震结构进行了单向及双向地震反应对比分析。结果表明，在单向和双向地震作用下，基础隔震结构的加速度反应和位移反应较为接近，但在双向地震作用下，支座的最大位移明显大于单向地震作用时的支座最大位移，因而在确定支座最大位移时应考虑双向地震作用的影响。     

Stochastic management of pump-and-treat strategies using surrogate functions

Typical pump-and-treat (PAT) optimization problems involve design of pumping schemes, while minimizing cost and meeting a set of constraints. Due to scarcity of information about the hydrogeological system, stochastic modeling approaches can be used to assess tradeoffs between optimality and reliability. Using a stochastic approach, the constrained, single-objective problem may be turned into a multiobjective problem by substituting constraint inequalities with an additional objective function (OF) that accounts for the reliability of the PAT process. In this work, two approaches are analyzed: in one case, the additional OF consists of the probability of failure of a given remediation policy; in another, the OF additional is represented by the recourse, namely the penalty cost induced by the violation of constraints. In order to overcome the overwhelming computational cost required by stochastic simulation, surrogate forms of the OFs are introduced. In the test case under investigation, such functions are estimated by a kriging interpolation of the OF over a series of data points obtained from stochastic simulations of flow and transport, and calibrated against stochastic optimization solutions. The analysis of the two approaches for addressing the tradeoff of cost vs. reliability indicates that recourse accounts not only for the frequency of constraint violations, as the probability of failure does, but also for the intensity with which these occur. Ultimately, the recourse method allows considering less restrictive policies, although these may be highly sensitive to the choice of penalty functions.     

Stochastic modelling and risk analysis of groundwater pollution using FORM coupled with automatic differentiation

The first order reliability method (FORM) has been widely used in probabilistic modelling of groundwater problems. The FORM approach requires an iterative optimization procedure to find out the system failure point (the most probable point).The advantages of this approach are that it does not require many computations in comparison with other methods when applied to simple problems, and it produces reasonably accurate results. However, it has been found that the computations of FORM can equal or exceed that of other methods in case of large number of variables.In this paper, a new implementation of FORM was proposed with more efficiency and accuracy than the traditional FORM method. In the proposed approach, automatic differentiation is used to obtain the gradient vector of the limit state function, which is required by FORM, instead of using finite difference estimation. This way, the first order derivative was obtained with a very good accuracy, and with less computational effort. Based on the obtained results, it is found that the proposed implementation of FORM is a very good tool for probabilistic risk assessment and uncertainty analysis in groundwater problems.