H13B-0402 1340h
Measuring a Small Hydraulic Gradient in the Presence of Noise
In naturally occurring flow systems the hydraulic gradient may be small, often less than 0.002- 0.001. These small gradients are hard to measure accurately at all scales. On regional flow maps, the accuracy of the head contour lines and the gradient is usually determined by the accuracy of the elevation of the top of the casing and of the well location on the regional map. These limitations on the regional scale accuracy will be improved in the future by the use of Global Positioning System (GPS) technology. However, for the present, some regional gradients and many local gradients - even those based on measurements made at local scales - are problematic in low gradient areas. This paper uses field data to demonstrate some of the problems associated with determining a small natural gradient in the vicinity of a research site, the Geohydrologic Experiment and Monitoring Site (GEMS) at the University of Kansas. The site is contained in an area of about 50 meters by 50 meters near the valley wall in the Kansas River valley north of Lawrence Kansas. The difficulty of determining the natural gradient was discovered about 10 years ago when attempting to design and run a bromide tracer test; the distances between wells at GEMS are too small (about 20m max.) to accurately determine the hydraulic gradient. The task is further complicated by the presence of rural water district wells some distance to the west of the site. Monitoring the water levels at the site reveals a noisy environment, primarily caused by the periodic pumping of the rural water district wells. In 2003 two additional wells were installed near the site, one to the east (96m) and one to the south (147m) of GEMS. With these larger distances between wells, monitored water level differences were more pronounced, permitting reliable gradient estimates to be calculated by accurately surveying the elevations and locations of the wells. Water levels were measured using accurately calibrated pressure transducers recording data at a high frequency over an extended period of time. The amplitude of the noise in the area was found to be larger than the head differences measured between wells; subsequently, techniques for extracting the usable signal were employed and found to be successful. This paper presents the results of our measurement of the natural gradient in the presence of noise, points out some of the pitfalls to avoid when measuring small gradients, and quantifies the uncertainties in flow velocity (magnitude and direction) that can result from gradient measurements in the presence of noise.
H13B-0403 1340h
The use of Bounded Intervals for Data Representation in the Evaluation of Uncertainties in Rainfall-Runoff Modelling
The evaluation of parameter or model structural uncertainty when considering modelling the Rainfall-Runoff process usually requires the consideration of at least two observational series, rainfall and stage. As well as being incommensurate with most rainfall-runoff models the observations are often made on a different spatial and/or temporal scale. Given this any `measure of association' between the observed data and data series used by the model is a representation of more than simply the observational uncertainty, but also any mapping resolving scale or commensurability differences. The realisation that the `measure of association' also represents a mapping can lead to an ill posed problem since the measure may be inseparable from the model for whose structural inadequacy it compensates. One method of achieving this separation is to select a priori a set of acceptable bounds for the `measures of association'. The work presented here considers a derivation of bounds for the `measures of association' for the Panola Mountain Research Watershed with reference to a simple hydraulic model. This model is then applied using a novel computational algorithm based upon particle filters to evaluate its parameter uncertainty.
H13B-0404 1340h
Representing nonstationarity in the rainfall-runoff relationship using a Hierarchical Mixture of Experts model
The evaluation and comparison of hydrological models has long been a challenge to the practicing hydrological community. With the variety of models available, modelers are faced with the problem of determining which is best for a particular modeling exercise. Here we present an alternative where, instead of choosing a single model, the catchment is allowed to dynamically exist in multiple hydrologic states. Each such state is represented by a unique rainfall-runoff model, and the "switch" from one state to the other occurs probabilistically depending on the catchment antecedent conditions. This new modeling framework, known as Hierarchical Mixture of Experts (HME), is applied to a number of Australian catchments having varying attributes. Results from this application are compared to the alternative where only one dominant hydrologic state is assumed to exist. The proposed alternatives are developed under a Bayesian framework, which is ideally suited here given the probabilistic basis of the HME model. We conclude by discussing how the HME modeling framework can be used for Predictions in Ungauged Basins (PUB). The traditional approach to PUB suffers from several drawbacks including the necessity of using a rigid modeling framework that can be easily parameterized and the requirement that the regionalization is applicable to the ungauged catchment it is intended for use in. We theorise instead that one may use an HME framework with a finite number of states to model multiple catchments, with the selection of each state depending on catchment characteristics and the modeled antecedent conditions.
H13B-0405 1340h
Simulated Streamflow Sensitivity due to Input and Parametric Uncertainty: Spatial Scaling
The influence of rainfall input and parametric uncertainty on the character of the flow simulation uncertainty is examined through ensemble simulations using a validated distributed hydrologic model. The study examines the sensitivity of ensemble flow simulations produced by the distributed model HRCDHM to uncertainty in parametric and radar rainfall input for two study watersheds in the southern Central Plains of the United States. The watersheds were included in the Distributed Model Intercomparison Project (DMIP) organized by the US National Weather Service Office of Hydrologic Development, and HRCDHM validated well in DMIP for both watersheds. Several scales of watershed resolution were employed for both study watersheds, and uncertainty scenarios included parametric uncertainty involving multiple soil model parameters simultaneously, and radar-rainfall uncertainty based on radar-pixel scale uncertainty scale to the subcatchment scale. Flow sensitivities were summarized in terms of a relative measure of the dispersion in the flow ensembles computed for selected events between May 1993 and July 1999. The results consistently show that the flow simulation uncertainty is strongly dependent on catchment scales, with significantly reduced flow sensitivity for larger watershed scales. The consistency of this result for the selected watershed locations may allow for the development of scaling relationships between catchment size and the flow uncertainty measure. Such a relationship potentially may be used to infer pronounced small-scale simulation uncertainties in distributed hydrologic model applications.
H13B-0406 1340h
A method for identifying sources of model uncertainty in rainfall-runoff simulations
A major goal in environmental modeling is identifying and quantifying sources of uncertainty in the modeling process. A forecast ensemble is developed in this study for a rainfall-runoff simulation system. This ensemble includes several quantitative precipitation estimates that serve as inputs to the Vflo` hydrologic model. The rainfall estimates are derived from rain gauges, radar, satellite, and combinations, and their probability distribution is assumed to encompass the true, but unknown, rainfall. Sensitive model parameters in the model are also perturbed within their physical bounds to create a combined input-parameter ensemble. If all major sources of uncertainty are accounted for, then observations of river discharge should fall within simulation bounds. Otherwise, there may be an additional error that lies within the model structure. Probability distributions derived from the forecast ensemble encompass streamflow observations for three hydrologic events examined during October and December on the Blue River Basin in Oklahoma. It is discovered, however, that all simulations from an ensemble created for a warm season case overforecast discharge peaks and volumes. Climatological rain gauge, discharge, and soil moisture observations are introduced to illuminate the source of uncertainty that was not accounted for in the combined input-parameter ensemble. Observations show a strong correlation between dry, deep-layer soils and significantly reduced runoff production (provided the same rainfall inputs) during the summer months. The Green and Ampt methodology is used in the model to compute soil infiltration rates. Evidence suggests additional abstractions such as interception by vegetation and deep cracks in the soil structure contribute to enhanced infiltration rates during the warm season. These effects need to be considered for future infiltration models.
H13B-0407 1340h
Inverse and forward modeling under uncertainty using MRE-based Bayesian approach
A stochastic inverse approach for subsurface characterization is proposed and applied to shallow vadose zone at a winery field site in north California and to a gas reservoir at the Ormen Lange field site in the North Sea. The approach is formulated in a Bayesian-stochastic framework, whereby the unknown parameters are identified in terms of their statistical moments or their probabilities. Instead of the traditional single-valued estimation /prediction provided by deterministic methods, the approach gives a probability distribution for an unknown parameter. This allows calculating the mean, the mode, and the confidence interval, which is useful for a rational treatment of uncertainty and its consequences. The approach also allows incorporating data of various types and different error levels, including measurements of state variables as well as information such as bounds on or statistical moments of the unknown parameters, which may represent prior information. To obtain minimally subjective prior probabilities required for the Bayesian approach, the principle of Minimum Relative Entropy (MRE) is employed. The approach is tested in field sites for flow parameters identification and soil moisture estimation in the vadose zone and for gas saturation estimation at great depth below the ocean floor. Results indicate the potential of coupling various types of field data within a MRE-based Bayesian formalism for improving the estimation of the parameters of interest.
H13B-0408 1340h
Uncertainty Analysis in Identifying the Source Location and Release History
In the real world, in case a high contaminant concentration plume is detected at monitoring wells, the source characters such as source location and release history are needed to be identified. The engineers need to evaluate an appropriate management or remediation strategy based on the information of source characters. The method of locating the source and determining the release history is an inverse problem and may be called as source identification. The groundwater contaminant source identification is an ill-posed problem since the contaminant fate and transport processes are irreversible. In this study, a method is developed using the simulated annealing (SA) incorporated with a two-dimensional transport model to determine the source characters containing the release magnitude and the release time. The proposed method minimizes the sum of square errors between the simulated concentrations and sampling concentrations at the monitoring wells. A hypothetical case is assumed to demonstrate the capability of the proposed method in identifying the contaminant source characters. Assume that a point source is injected instantaneously into the aquifer twice at unknown location and times. With the sampling concentrations, the proposed method is used to determine the source characters. The hydrogeologic parameters such as groundwater velocity and dispersion coefficient are the essential information in the work of source identification. The main objective of this study is to perform the sensitivity analysis for the parameters of seepage velocity and dispersion coefficient in identifying the source location and release history. In addition, different values of initial temperature are also used to examine the stability of SA. The results of sensitivity analysis for the initial temperature indicate that SA yields almost the same identification results when using different initial temperatures. In the sensitivity analysis for the parameters of seepage velocity and dispersion coefficient, the source location is more insensitive to the change of longitudinal dispersion coefficient than those of the seepage velocity and transverse dispersion coefficient. However, the release history is very sensitivity to the changes of longitudinal and transverse dispersion coefficients due to the fact that conceptually the dispersion spreads the contaminant in longitudinal and transverse directions at the advection front. In short, the results show that the estimated source location and release history are more sensitivity to the variation of dispersion coefficient than that of groundwater velocity.
H13B-0409 1340h
Combined States and Parameters Estimation in Transient Groundwater Modeling by the Simulated Based Particle Filtering
Transient modeling of groundwater flow and transport is done in a sequential pattern, and modelers make the predictions of the system states over a series of discrete time points. The nonlinearity of the system dynamic and sparse knowledge of the model parameters make such a task difficult, and the uncertainty about the spatial distributed parameters, such as hydraulic conductivity, dispersitivity, make it even more challenging. Generally the prior knowledge about the model parameters is initially used in modeling, but the sequential observations of system states can also be utilized to improve the predictions and decrease the uncertainty of the model parameters. Such observations are usually available in the real situations. The objective of this research is to introduce the simulated-based particle filtering into groundwater modeling and inverse modeling. By sequentially sampling the states predictions and model parameters conditioning in observation, the statistic of the dynamic states and the model parameters can be evaluated simultaneously, without assumption of normal distribution and linearization of differential equation of model dynamics. Its applications in the lumped and multi-dimensional modeling of groundwater flow and transport are explored.
H13B-0410 1340h
Long-term Monitoring Algorithm Verification Using an Intermediate Scale Groundwater Facility
Long term groundwater monitoring networks created by nonlinear optimization methods coupled with stochastic transport simulators are currently available. However, few of the schemes have been tested. In an effort to test one of these design algorithms a 4.22 by 2.74 by 2.13 meter intermediate-scale indoor facility that mimics a heterogeneous subsurface environment was constructed at the University of Vermont. Time Domain Reflectometry (TDR) sensors were installed in the facility at 105 locations to monitor the change of electrical conductivities in the subsurface. A 20-day point source continuous-injection experiment was conducted. Salt concentrations were collected at a frequency of 20 minutes per measurement per sensor. An exhaustive dataset of a migrating plume was thereby established. Various tests were conducted to provide the measures and statistics of the hydraulic conductivities (K) in the subsurface. Latin Hypercube Sampling based random field generator was adopted to create the realizations of random fields of hydraulic conductivity based on the information of K from the previous step. Groundwater flow and transport simulators were used to calculate the associated concentration fields. A static Kalman filter and a genetic algorithm were then used to choose sampling locations and times based on minimizing the cost subject to the constraint of a desired accuracy. A monitoring network was thereby designed. Computed concentration plume statistics were updated using the information obtained from samples proposed by the monitoring network. Finally, these updated plumes were compared with the actual plumes to determine the effectiveness of the optimized monitoring network.
H13B-0411 1340h
Search Strategy for a DNAPL Source
The plume emanating from a DNAPL source is typically quite large and easily discovered as opposed to the identification of the location of a DNAPL source, which can be a very difficult task because it is a small target. The goal of this work is to identify the source of DNAPL contamination using an optimal search algorithm which exploits the above observation. The target locations of the possible sources are identified and given initial weights using an approach called information fusion. In this approach each possible source location is described by an n-dimensional vector, whose coordinates are values of identifying features of the source, such as its geography. Each feature value is compared with some prototype value, which gives a degree of confidence of the statement `source location i belongs to the group of true source locations'. Using a fuzzy integral all the individual degrees of confidence are combined and a global degree of confidence (weight) is assigned to each possible source. Given the initial identification of the sources the overall, the strategy uses stochastic groundwater flow and transport modeling under the assumption that hydraulic conductivity is known with uncertainty (Monte Carlo approach). The hydraulic conductivity realizations are obtained using the Latin hypercube sampling strategy. The algorithm defines how to achieve an acceptable level of source-location accuracy with the least possible number of water quality samples. Each new concentration sample selected is the one that reduces the total concentration variance the most. After each sample is taken the plume is updated using a Kalman Filter. The plumes emanating from each individual source are calculated using the Monte Carlo approach and are compared with the updated plume. The scores obtained from this comparison are used as input weights for each individual source, and the above steps are repeated until the optimal source location is found.
H13B-0412 1340h
Integration of Fuzzy and Probabilistic Information in the Description of Hydraulic Conductivity
Evaluation of the heterogeneity of hydraulic conductivity, K, is a well-known problem in groundwater hydrology. The open question is how to fully represent a given highly heterogeneous K field and its inherent uncertainty at least cost. Today, most K fields are analyzed using field test data and probability theory. Uncertainty is usually reported in the spatial covariance. In an attempt to develop a more cost effective method which still provides an accurate approximation of a K field, we propose using an evidence theory framework to merge probabilistic and fuzzy (or possibilistic) information in an effort to improve our ability to fully define a K field. The tool chosen to fuse probabilistic information obtained via experiment and subjective information provided by the groundwater professional is Dempster's Rule of Combination. In using this theory we must create mass assignments for our subject of interest, describing the degree of evidence that supports the presence of our subject in a particular set. These mass assignments can be created directly from the probabilistic information and, in the case of the subjective information, from feedback we obtain from an expert. The fusion of these two types of information provides a better description of uncertainty than would typically be available with just probability theory alone.
H13B-0413 1340h
Characterizing the K-field in a Spatial Domain Using Fuzzy and Case-Based Spatial Reasoning
Kriging is widely used for the estimation of spatially distributed variables such as hydraulic conductivity. Relying on given measurements of hydraulic conductivity, the kriging system interpolates values to any location in the domain with minimal estimation error. However, the kriging system fails to respect the variable's true spatial dispersion, as it generally produces a smoother-than-reality contour map of values. Furthermore, the estimations of values in areas where few measurements exist are highly suspect. A promising alternative to kriging is the application of expert knowledge in the form of fuzzy rules. With an understanding of the available measurement data, an expert may be able to make statements related to the variable, using fuzzy terms such as {\bf near}, {\bf about 5 meters}, and {\bf roughly 10 m/day}. In the case of hydraulic conductivity the fuzzy rules may tap into the knowledge that experts possess regarding {\it spatial relations} amongst geologic entities. For example, a typical statement may be {\it If a body of clay exists in the northeast section of the domain, then there exists a silty-clay formation about 5 meters to the southwest.} The expert may also describe {\it hydrological relations} between soil types and hydraulic conductivity values such as in the expression {\it If soil type is clay, then hydraulic conductivity is low.} A case-based reasoning system may be used to supplement the expert statements by adding information that cannot be captured by expert knowledge. In this approach a fuzzy retrieval algorithm in the system would find previously studied and documented spatial domains similar to the domain of interest with the intent of harvesting information concerning the possible geologic make-up and hydraulic conductivity field of our spatial domain. Thus, while kriging is restricted to the available measurements, the expression of expert knowledge through fuzzy rules and the information from previous cases supplements the {\it a priori} data to provide an enhanced representation of the field.
H13B-0414 1340h
Uncertainty in Computational Simulations of Geophysical Mass Flows
We address the uncertainty inherent in modeling and computational simulations of geophysical mass flows. This uncertainty arises from unresolved physics, modeling simplifications, errors in terrain data or constitutive parameters, and the inaccuracies of any numerical method. The errors must be incorporated directly into the modeling and computations, and ensembles of solutions to the deterministic model equations, coupled to statistical analysis, are necessary to provide meaningful results for hazard assessment and risk mitigation associated with geophysical mass flows such as avalanches and landslides. In recent years a set of depth averaged equations (the Savage-Hutter model) with simple constitutive modeling assumptions has come into wide usage. In earlier work we developed the TITAN toolset that uses state of the art numerical methodology (high performance computing, adaptive gridding, etc.) to construct approximate solutions to these systems of equations for modeling flow over natural terrain. We describe in this contribution our efforts at incorporating uncertainty representations into this toolset. We use the recently developed methodology of polynomial chaos to represent uncertainty in the outputs based on input data uncertainty. In applying the polynomial chaos methodology to such systems we have had to overcome a series of technical difficulties. We will describe our solutions to each of these obstacles. Real-world results showing the comparison between model outputs and data collected in the field will be used in illustration.
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