Stochastic models in biology.

Over the past 50 years, humans have altered their environment to a significant extent, although human well-being is dependent on ecosystem functioning. Ecosystems are particularly affected by unsustainable use of resources, such as, food, water, and timber. Ecosystem functions depend on water, carbon, and other nutrients cycles. Human activities have modified these cycles in a number of way. Use of ecosystems for recreation, spiritual enrichment, cultural purposes, and for other short term benefits is growing continuously, although ecosystem capacity to provide such services has reported to be declined significantly. Human well-being depends on material welfare, health, good social relations, security and freedom, which are affected by changes in ecosystem services. Intensive ecosystem use often produces short-term advantage. Poverty level remains high in more than one billion people, who are dependent on ecosystems with an income of less than $1 per day as reported elsewhere. Regions including some parts of Africa, Asia and Latin America have the greatest ecosystemrelated problems and are facing developmental challenges. Habitat change occurs, for instance, when the area of land used for agriculture or cities is expanded. Instability and unproductivity including desertification, water logging, mineralization and many other unwanted outcomes throughout the world are continuing. Habitat fragmentation by roads, canals, power lines limits the species potential for dispersal and colonization. Indirect drivers, like changes in human population, economic activity and technology as well as socio-political and cultural factors affect ecosystems by influencing direct drivers. World climate has changed and continues to change, affecting temperature, rainfall and sea levels. Intensive fertilizer use has polluted ecosystems. Climate change and high nutrient levels in water are becoming increasing problems. Ecosystem management for shortterm benefits is increasing. Loss of biodiversity makes it difficult for ecosystems to recover from damage. Once an ecosystem has undergone an abrupt change, recovery to the original state is slow, costly, and sometimes impossible. Changes in ecosystems complexity- functioning relationships could diminish the stability, resistance and resilience of managed terrestrial ecosystems, and may jeopardize important food and fibre sources, and ability of natural ecosystems both to provide natural resources, and to remove pollutants from atmosphere. Ecological complexity and ecosystem functioning depend on factors that govern species coexistence. Complexity of landscapes is determined by number of ecosystem types, their characteristics, their sizes and shapes, and associated connectivity. Complexity at this scale would have large consequences on regional to global scale processes. Presence and arrangement of keystone ecosystem types, such as, wetlands often determine total carbon and nitrogen balance of a region. Changes in average or extreme environmental events and intense land use management are believed to increase species extinction rate in isolated habitat fragments. Loss of key species, such as, top predators, fruit dispersers and pollinators from habitat may severely disrupt ecosystems functioning. Land use changes due to expanding urbanization, concomitant landscape fragmentation and intensification of production systems. Such change results in transformation of an ecosystem, form one state to another state, via a transition phase. The combined value of 17 ecosystem services has been reported in the estimated range of US$16-54 trillion per year by Costanza and others. About 30% of modern medicines are developed from plants and animals, and 10 of the world's 25 topselling drugs in 1997 were reported to be derived from natural sources. Global market value of pharmaceuticals derived from genetic resources is estimated at US $ 75 000-150 000 million annually. Some 75% of the world's populations rely for health care on traditional medicines, which are derived directly from natural sources as recorded elsewhere. Socio-economic development of human civilization and human well-being depends on long-term health of environment including ecosystems. Environmental problems are generally addressed in isolation, but practically such problems are interrelated, and originate from the root cause of unsustainable development. Damage to natural ecosystems and release of environmental pollutants must be minimized for protecting natural ecosystem, and human well-being.

Ecosystems and Human Well-Being

Geophysics provides a multidimensional suite of investigative methods that are transforming our ability to see into the very fabric of the subsurface environment, and monitor the dynamics of its fluids and the biogeochemical reactions that occur within it. Here we document how geophysical methods have emerged as valuable tools for investigating shallow subsurface processes over the past two decades and offer a vision for future developments relevant to hydrology and also ecosystem science. The field of “hydrogeophysics” arose in the late 1990s, prompted, in part, by the wealth of studies on stochastic subsurface hydrology that argued for better field-based investigative techniques. These new hydrogeophysical approaches benefited from the emergence of practical and robust data inversion techniques, in many cases with a view to quantify shallow subsurface heterogeneity and the associated dynamics of subsurface fluids. Furthermore, the need for quantitative characterization stimulated a wealth of new investigations into petrophysical relationships that link hydrologically relevant properties to measurable geophysical parameters. Development of time-lapse approaches provided a new suite of tools for hydrological investigation, enhanced further with the realization that some geophysical properties may be sensitive to biogeochemical transformations in the subsurface environment, thus opening up the new field of “biogeophysics.” Early hydrogeophysical studies often concentrated on relatively small “plot-scale” experiments. More recently, however, the translation to larger-scale characterization has been the focus of a number of studies. Geophysical technologies continue to develop, driven, in part, by the increasing need to understand and quantify key processes controlling sustainable water resources and ecosystem services.

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The emergence of hydrogeophysics for improved understanding of subsurface processes over multiple scales.

Groundwater simulation models have been incorporated into a genetic algorithm to solve three groundwater management problems: maximum pumping from an aquifer; minimum cost water supply development; and minimum cost aquifer remediation. The results show that genetic algorithms can effectively and efficiently be used to obtain globally (or, at least near globally) optimal solutions to these groundwater management problems. The formulation of the method is straightforward and provides solutions which are as good as or better than those obtained by linear and nonlinear programming. Constraints can be incorporated into the formulation and do not require derivatives with respect to decision variables as in nonlinear programming. More complicated problems, such as transient pumping and multiphase remediation, can be formulated and solved using this method. The computational time required for the solution of genetic algorithm groundwater management models increases with the complexity of the problem. The speedup attainable by solving genetic algorithm problems on massively parallel computers is significant for problems where the simulation time required to complete each generation is high. 47 refs., 9 figs., 5 tabs.

Genetic algorithm solution of groundwater management models

A new methodology for solution of the inverse problem in groundwater hydrology is proposed and applied to a site in southeastern New Mexico with extensive hydrogeologic data. The methodology addresses the issue of nonuniqueness of the inverse solutions by generating an ensemble of transmissivity fields considered to be equally likely, each of which is in agreement with the measured transmissivity and pressure data. It consists of generating a selected number of conditionally simulated transmissivity fields and then calibrating each of the fields to match the measured steady state or transient pressures, in a least squares sense. The calibration phase involves an iterative implementation of an automated pilot point approach coupled with conditional simulations. Pilot points are the parameters of calibration. They are synthetic transmissivity data which are added to the transmissivity database to produce a revised conditional simulation during calibration. Coupled kriging and adjoint sensitivity analysis is employed for the optimal location of pilot points, and gradient search methods are used to derive their optimal transmissivities. The pilot point methodology is well suited for characterizing the spatial variability of the transmissivity field in contrast to methods using zonation. Pilot points are located where their potential for minimizing the objective function is the highest. This minimizes the perturbations in the transmissivities which are optimally assigned to the pilot point and results in minimal changes to the covariance structure of the transmissivity field. The calibrated fields honor the transmissivity measurements at their locations, preserve the variogram, and match the measured pressures in a least squares sense. Since there are numerous options in the execution of this methodology, computational experiments have been conducted to identify the most efficient among them. The method has been applied to the Waste Isolation Pilot Plant (WIPP) site, in southeastern New Mexico, where the U.S. Department of Energy is conducting probabilistic system assessment for the permanent disposal of transuranic nuclear waste. The resulting calibrated transmissivity fields are input to a Monte Carlo analysis of the total system performance. The present paper, paper 1 of a two-paper presentation, describes the methodology. Paper 2, a companion paper, presents the methodology's application to the WIPP site.

Pilot Point Methodology for Automated Calibration of an Ensemble of conditionally Simulated Transmissivity Fields: 1. Theory and Computational Experiments

As a justification for the geoelectric characterization of the hydraulic conductivity field, this paper shows theoretically and empirically that electrical and hydraulic (eh) conductivities of aquifers can be correlated. The correlation, demonstrated at the microscale by a published network model of eh transport, arises from the fact that both eh conductivities are a function of connected pore volumes and connected pore surface areas. By considering skewed pore size distributions the microscale equations of eh conductivity scale up to power laws of porosity and specific surface area similar to Archie's law and the Kozeny equation. Also, a third, apparently unreported Archie-type power law relating electrical conductivity to specific surface area and the cation exchange phenomenon is predicted theoretically and confirmed experimentally. These equations imply a simple log-log linear correlation between eh conductivities that is either positive or negative. The positive correlation corresponds to a pore-volume-dominated electrical flow environment and the negative correlation corresponds to a pore-surface-dominated electrical flow environment. These relationships are supported by many published laboratory and field investigations cited in the paper.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2000WR900165

On the electrical‐hydraulic conductivity correlation in aquifers

The general formulation of the environmental risk problem captures the entire process of identifying the source term of the risk agent, its fate and transport through porous media, estimation of human exposure, and conversion of such exposure into the risk level. The contaminant fate and transport is modeled using the solute flux formulation evaluated with its first two moments, which explicitly account for the spatial variability of the velocity field, sorption properties, and parametric uncertainty through the first-order analysis. The risk level is quantified on the basis of carcinogenicity using the risk factor (which describes the risk per unit dose or unit intake) employed to the total doses for individuals potentially consuming radionuclide-contaminated groundwater. As a result of the probabilistic formulation in the solute flux and uncertainty in the water intake and dose-response functions, the total risk level is expressed as a distribution rather than a single estimate. The results indicate that the geologic heterogeneity and uncertainty in the sorption estimate are the two most important factors for the risk evaluation from the physical and chemical processes, while the mean risk factor is a crucial parameter in the risk formulation.

Evaluation of Risk from Contaminants Migrating by Groundwater

A common method of aquifer decontamination is through “pump and treat” systems. Mathematical models of flow and transport can be incorporated into an optimization scheme to determine the most cost-effective pumping pattern and schedule, provided that the parameters of the system are somehow known. In practice, however, there is seldom enough information to specify with certainty all the parameters of the system. Parameters are obtained from measurements through inverse modeling and are subject to error. This work presents an optimization methodology which accounts for and reduces parameter uncertainty. It is shown that in an average sense the cost consists of two parts: one which depends only on the best estimates and another which increases with the error in parameter estimation. A policy which minimizes the sum of the two parts is developed. The proposed methodology elucidates the dual control aspect of stochastic control, i.e., the control policy (pumping schedule) optimizes the objective function by, among other things, improving the accuracy of estimation of system parameters (i.e., dispersivity, transmissivity). For purposes of illustration, the proposed method is applied to a simple case. Comparison with an optimization method which neglects uncertainty in parameter estimation shows the superiority of the proposed algorithm.

Optimization of the pumping schedule in aquifer remediation under uncertainty

[1] In this paper we present flow and travel time ensemble statistics based on a new simulation methodology, the adaptive Fup Monte Carlo method (AFMCM). As a benchmark case, we considered two-dimensional steady flow in a rectangular domain characterized by multi-Gaussian heterogeneity structure with an isotropic exponential correlation and lnK variance σY2 up to 8. Advective transport is investigated using the travel time framework where Lagrangian variables (e.g., velocity, transverse displacement, or travel time) depend on space rather than on time. We find that Eulerian and Lagrangian velocity distributions diverge for increasing lnK variance due to enhanced channeling. Transverse displacement is a nonnormal for all σY2 and control planes close to the injection area, but after xIY = 20 was found to be nearly normal even for high σY2. Travel time distribution deviates from the Fickian model for large lnK variance and exhibits increasing skewness and a power law tail for large lnK variance, the slope of which decreases for increasing distance from the source; no anomalous features are found. Second moment of advective transport is analyzed with respect to the covariance of two Lagrangian velocity variables: slowness and slope which are directly related to the travel time and transverse displacement variance, which are subsequently related to the longitudinal and transverse dispersion. We provide simple estimators for the Eulerian velocity variance, travel time variance, slowness, and longitudinal dispersivity as a practical contribution of this analysis. Both two-parameter models considered (the advection-dispersion equation and the lognormal model) provide relatively poor representations of the initial part of the travel time probability density function in highly heterogeneous porous media. We identify the need for further theoretical and experimental scrutiny of early arrival times, and the need for computing higher-order moments for a more accurate characterization of the travel time probability density function. A brief discussion is presented on the challenges and extensions for which AFMCM is suggested as a suitable approach.

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Flow and travel time statistics in highly heterogeneous porous media

The relative dispersion framework for the non-reactive and reactive solute flux in aquifers is presented in terms of the first two statistical moments. The solute flux is described as a space time process where time refers to the solute flux breakthrough and space refers to the transverse displacement distribution at the control plane. The statistics of the solute flux breakthrough and transversal displacement distributions are derived by analysing the motion of particle pairs. The results indicate that the relative dispersion formulation approaches the absolute dispersion results on increasing the source size (e.g. > 10 heterogeneity scales). The solute flux statistics, when sampling volume is accounted for, show a flattened distribution for the solute flux variance in the space-time domain. For reactive solutes, the solute flux shows a tailing phenomenon in time while solute flux variance exhibits bi-modality in transverse distribution during the recession stage of the solute breakthrough. The solute flux correlation structure is defined as an integral measure over space and time, providing a potentially useful tool for sampling design in the subsurface.

Roko Andričević

Papers

On the electrical‐hydraulic conductivity correlation in aquifers

Evaluation of Risk from Contaminants Migrating by Groundwater

Optimization of the pumping schedule in aquifer remediation under uncertainty

Flow and travel time statistics in highly heterogeneous porous media

Relative dispersion for solute flux in aquifers