Direct Computation of Stochastic Flow in Reservoirs with Uncertain Parameters

A direct method is presented for determining the uncertainty in reservoir pressure, flow, and net present value (NPV) using the time-dependent, one phase, two- or three-dimensional equations of flow through a porous medium. The uncertainty in the solution is modelled as a probability distribution function and is computed from given statistical data for input parameters such as permeability. The method generates an expansion for the mean of the pressure about a deterministic solution to the system equations using a perturbation to the mean of the input parameters. Hierarchical equations that define approximations to the mean solution at each point and to the field covariance of the pressure are developed and solved numerically. The procedure is then used to find the statistics of the flow and the risked value of the field, defined by the NPV, for a given development scenario. This method involves only one (albeit complicated) solution of the equations and contrasts with the more usual Monte-Carlo approach where many such solutions are required. The procedure is applied easily to other physical systems modelled by linear or nonlinear partial differential equations with uncertain data.

Calibration of uncertain flood inundation models using remotely sensed water levels

A traditional method of validating the performance of a flood model when remotely sensed data of the flood extent are available is to compare the predicted flood extent to that observed. The performance measure employed often uses areal pattern-matching to assess the degree to which the two extents overlap. Recently, remote sensing of flood extents using synthetic aperture radar (SAR) and airborne scanning laser altimetry (LIDAR) has made more straightforward the synoptic measurement of water surface elevations along flood waterlines, and this has emphasised the possibility of using alternative performance measures based on height. This paper considers the advantages that can accrue from using a performance measure based on waterline elevations rather than one based on areal patterns of wet and dry pixels. The two measures were compared for their ability to estimate flood inundation uncertainty maps from a set of model runs carried out to span the acceptable model parameter range in a GLUE-based analysis. A 1 in 5-year flood on the Thames in 1992 was used as a test event. As is typical for UK floods, only a single SAR image of observed flood extent was available for model calibration and validation. A simple implementation of a two-dimensional flood model (LISFLOOD-FP) was used to generate model flood extents for comparison with that observed. The performance measure based on height differences of corresponding points along the observed and modelled waterlines was found to be significantly more sensitive to the channel friction parameter than the measure based on areal patterns of flood extent. The former was able to restrict the parameter range of acceptable model runs and hence reduce the number of runs necessary to generate an inundation uncertainty map. A result of this was that there was less uncertainty in the final flood risk map. The uncertainty analysis included the effects of uncertainties in the observed flood extent as well as in model parameters. The height-based measure was found to be more sensitive when increased heighting accuracy was achieved by requiring that observed waterline heights varied slowly along the reach. The technique allows for the decomposition of the reach into sections, with different effective channel friction parameters used in different sections, which in this case resulted in lower r.m.s. height differences between observed and modelled waterlines than those achieved by runs using a single friction parameter for the whole reach. However, a validation of the modelled inundation uncertainty using the calibration event showed a significant difference between the uncertainty map and the observed flood extent. While this was true for both measures, the difference was especially significant for the height-based one. This is likely to be due to the conceptually simple flood inundation model and the coarse application resolution employed in this case. The increased sensitivity of the height-based measure may lead to an increased onus being placed on the model developer in the production of a valid model

Variational data assimilation for parameter estimation: application to a simple morphodynamic model

Data assimilation is a sophisticated mathematical technique for combining observational data with model predictions to produce state and parameter estimates that most accurately approximate the current and future states of the true system. The technique is commonly used in atmospheric and oceanic modelling, combining empirical observations with model predictions to produce more accurate and well-calibrated forecasts. Here, we consider a novel application within a coastal environment and describe how the method can also be used to deliver improved estimates of uncertain morphodynamic model parameters. This is achieved using a technique known as state augmentation. Earlier applications of state augmentation have typically employed the 4D-Var, Kalman filter or ensemble Kalman filter assimilation schemes. Our new method is based on a computationally inexpensive 3D-Var scheme, where the specification of the error covariance matrices is crucial for success. A simple 1D model of bed-form propagation is used to demonstrate the method. The scheme is capable of recovering near-perfect parameter values and, therefore, improves the capability of our model to predict future bathymetry. Such positive results suggest the potential for application to more complex morphodynamic models.

Over-parameterised, uncertain 'mathematical marionettes' - How can we best use catchment water quality models? An example of an 80-year catchment-scale nutrient balance

The Integrated Catchment Model of Nitrogen (INCA-N) was applied to the River Lambourn, a Chalk river-system in southern England. The model's abilities to simulate the long-term trend and seasonal patterns in observed stream water nitrate concentrations from 1920 to 2003 were tested. This is the first time a semi-distributed, daily time-step model has been applied to simulate such a long time period and then used to calculate detailed catchment nutrient budgets which span the conversion of pasture to arable during the late 1930s and 1940s. Thus, this work goes beyond source apportionment and looks to demonstrate how such simulations can be used to assess the state of the catchment and develop an understanding of system behaviour. The mass-balance results from 1921, 1922, 1991, 2001 and 2002 are presented and those for 1991 are compared to other modelled and literature values of loads associated with nitrogen soil processes and export. The variations highlighted the problem of comparing modelled fluxes with point measurements but proved useful for identifying the most poorly understood inputs and processes thereby providing an assessment of input data and model structural uncertainty. The modelled terrestrial and instream mass-balances also highlight the importance of the hydrological conditions in pollutant transport. Between 1922 and 2002, increased inputs of nitrogen from fertiliser, livestock and deposition have altered the nitrogen balance with a shift from possible reduction in soil fertility but little environmental impact in 1922, to a situation of nitrogen accumulation in the soil, groundwater and instream biota in 2002. In 1922 and 2002 it was estimated that approximately 2 and 18 kg N ha(-1) yr(-1) respectively were exported from the land to the stream. The utility of the approach and further considerations for the best use of models are discussed. (C) 2008 Elsevier B.V. All rights reserved.

Single versus multiple occupancy - Effects on toxicity parameters measured on Eisenia fetida in lead nitrate-treated soil

The mortality (7 and 14 d), weight change (7 and 14 d), and metal uptake of Eisenia fetida (Savigny, 1826) kept in Pb(NO3)(2)-treated Kettering loam soil in single- and multiple-occupancy (10 earthworms) test containers were determined. The number of earthworms to dry mass (g) ratio of soil was 1:50 in both sets of test containers. Lead concentrations were in the nominal range of 0 to 10,000 mg Pb/kg soil (mg/kg hereafter). Levels of mortality at a given concentration were statistically identical between the single- and multiple-occupancy tests, except at 1,800 mg/kg, at which significantly (p less than or equal to 0.05) more mortality occurred in the multiple-occupancy tests. Death of individual earthworms in the multiple-occupancy tests did not trigger death of the other earthworms in that soil. The LC50 values (concentration statistically likely to kill 50% of the population) were identical between the multiple- and single-occupancy soils: 2,662 mg/kg (2,598-2,984, 7 d) and 2,589 mg/kg (2,251-3,013, 14 d) for the multiple-occupancy soils and 2,827 mg/kg (2,443-3,168, both 7 and 14 d) for the single-occupancy soils (values in brackets represent the 95% confidence intervals). Data were insufficient to calculate the concentration statistically likely to reduce individual earthworm mass by 50% (EC50), but after 14 d, the decrease in earthworm weight in the 1,800 and 3,000 mg/kg tests was significantly greater in the multiple- than in the single-occupancy soils. At 1,000, 1,800, and 3,000 mg/kg tests, earthworm Pb tissue concentration was significantly (p less than or equal to 0.05) greater in earthworms from the multiple-occupancy soils. The presence of earthworms increased the NH3 content of the soil; earthworm mortality increased NH3 concentrations further but not to toxic levels.

Identifying the critical parameters of a cyanobacterial growth and movement model by using generalised sensitivity analysis

Bloom-forming and toxin-producing cyanobacteria remain a persistent nuisance across the world. Modelling of cyanobacteria in freshwaters is an important tool for understanding their population dynamics and predicting the location and timing of the bloom events in lakes and rivers. A new deterministic-mathematical model was developed, which simulates the growth and movement of cyanobacterial blooms in river systems. The model focuses on the mathematical description of the bloom formation, vertical migration and lateral transport of colonies within river environments by taking into account the major factors that affect the cyanobacterial bloom formation in rivers including, light, nutrients and temperature. A technique called generalised sensitivity analysis was applied to the model to identify the critical parameter uncertainties in the model and investigates the interaction between the chosen parameters of the model. The result of the analysis suggested that 8 out of 12 parameters were significant in obtaining the observed cyanobacterial behaviour in a simulation. It was found that there was a high degree of correlation between the half-saturation rate constants used in the model.

Effects of metals on life cycle parameters of the earthworm Eisenia fetida exposed to field-contaminated, metal-polluted soils

Two control and eight field-contaminated, metal-polluted soils were inoculated with Eisenia fetida (Savigny, 1826). Three, 7, 14, 21, 28 and 42 days after inoculation, earthworm survival, body weight, cocoon production and hatching rate were measured. Seventeen metals were analysed in E.fetida tissue, bulk soil and soil solution. Soil organic carbon content, texture, pH and cation exchange capacity were also measured. Cocoon production and hatching rate were more sensitive to adverse conditions than survival or weight change. Soil properties other than metal concentration impacted toxicity. The most toxic soils were organic-poor (1-10 g C kg(-1)), sandy soils (c. 74% sand), with intermediate metal concentrations (e.g. 7150-13, 100 mg Ph kg(-1), 2970-53,400 mg Zn kg(-1)). Significant relationships between soil properties and the life cycle parameters were determined. The best coefficients of correlation were generally found for texture, pH, Ag, Cd, Mg, Pb, Tl, and Zn both singularly and in multivariate regressions. Studies that use metal-amended artificial soils are not useful to predict toxicity of field multi-contaminated soils. (c) 2007 Elsevier Ltd. All rights reserved.

An application of the GLUE methodology for estimating the parameters of the INCA-N model

The conceptual and parameter uncertainty of the semi-distributed INCA-N (Integrated Nutrients in Catchments-Nitrogen) model was studied using the GLUE (Generalized Likelihood Uncertainty Estimation) methodology combined with quantitative experimental knowledge, the concept known as 'soft data'. Cumulative inorganic N leaching, annual plant N uptake and annual mineralization proved to be useful soft data to constrain the parameter space. The INCA-N model was able to simulate the seasonal and inter-annual variations in the stream-water nitrate concentrations, although the lowest concentrations during the growing season were not reproduced. This suggested that there were some retention processes or losses either in peatland/wetland areas or in the river which were not included in the INCA-N model. The results of the study suggested that soft data was a way to reduce parameter equifinality, and that the calibration and testing of distributed hydrological and nutrient leaching models should be based both on runoff and/or nutrient concentration data and the qualitative knowledge of experimentalist. (c) 2006 Elsevier B.V. All rights reserved.

Over-parameterised, uncertain 'mathematical marionettes' - How can we best use catchment water quality models? An example of an 80-year catchment-scale nutrient balance

The Integrated Catchment Model of Nitrogen (INCA-N) was applied to the River Lambourn, a Chalk river-system in southern England. The model's abilities to simulate the long-term trend and seasonal patterns in observed stream water nitrate concentrations from 1920 to 2003 were tested. This is the first time a semi-distributed, daily time-step model has been applied to simulate such a long time period and then used to calculate detailed catchment nutrient budgets which span the conversion of pasture to arable during the late 1930s and 1940s. Thus, this work goes beyond source apportionment and looks to demonstrate how such simulations can be used to assess the state of the catchment and develop an understanding of system behaviour. The mass-balance results from 1921, 1922, 1991, 2001 and 2002 are presented and those for 1991 are compared to other modelled and literature values of loads associated with nitrogen soil processes and export. The variations highlighted the problem of comparing modelled fluxes with point measurements but proved useful for identifying the most poorly understood inputs and processes thereby providing an assessment of input data and model structural uncertainty. The modelled terrestrial and instream mass-balances also highlight the importance of the hydrological conditions in pollutant transport. Between 1922 and 2002, increased inputs of nitrogen from fertiliser, livestock and deposition have altered the nitrogen balance with a shift from possible reduction in soil fertility but little environmental impact in 1922, to a situation of nitrogen accumulation in the soil, groundwater and instream biota in 2002. In 1922 and 2002 it was estimated that approximately 2 and 18 kg N ha(-1) yr(-1) respectively were exported from the land to the stream. The utility of the approach and further considerations for the best use of models are discussed. (C) 2008 Elsevier B.V. All rights reserved.

Robust dynamics of Amazon dieback to climate change with perturbed ecosystem model parameters

Climate change science is increasingly concerned with methods for managing and integrating sources of uncertainty from emission storylines, climate model projections, and ecosystem model parameterizations. In tropical ecosystems, regional climate projections and modeled ecosystem responses vary greatly, leading to a significant source of uncertainty in global biogeochemical accounting and possible future climate feedbacks. Here, we combine an ensemble of IPCC-AR4 climate change projections for the Amazon Basin (eight general circulation models) with alternative ecosystem parameter sets for the dynamic global vegetation model, LPJmL. We evaluate LPJmL simulations of carbon stocks and fluxes against flux tower and aboveground biomass datasets for individual sites and the entire basin. Variability in LPJmL model sensitivity to future climate change is primarily related to light and water limitations through biochemical and water-balance-related parameters. Temperature-dependent parameters related to plant respiration and photosynthesis appear to be less important than vegetation dynamics (and their parameters) for determining the magnitude of ecosystem response to climate change. Variance partitioning approaches reveal that relationships between uncertainty from ecosystem dynamics and climate projections are dependent on geographic location and the targeted ecosystem process. Parameter uncertainty from the LPJmL model does not affect the trajectory of ecosystem response for a given climate change scenario and the primary source of uncertainty for Amazon 'dieback' results from the uncertainty among climate projections. Our approach for describing uncertainty is applicable for informing and prioritizing policy options related to mitigation and adaptation where long-term investments are required.