Null-space Monte Carlo Method Research Articles

Predictive uncertainty analysis for a highly parameterized karst aquifer using null-space Monte Carlo

Inverse problems in hydrogeology pose a great challenge for modelers as they are ill-posed, resulting in a non-unique solution. High computational resources are needed for the calibration process, especially in the case of highly parameterized aquifers like karst limestone, characterized by significant heterogeneity. The null-space Monte Carlo (NSMC) is a parameter-constrained Monte Carlo approach that can be used to quantify uncertainty, as it produces a set of solutions that calibrate the model. This method is used to assess uncertainty in the calibration of a karst aquifer in Qatar, which has high heterogeneity. Pilot points were used to reflect the geostatistics of the calibrated field, and the calibration results at these points were interpolated over the aquifer area using kriging. The NSMC was then used to produce 200 realizations of the null-space parameter field using the constrained random variable of hydraulic conductivity. The null-space realizations were then incorporated into the parameter space derived from the calibrated model. Statistical analysis of the calibrated hydraulic conductivity revealed a variation ranging from 0.1 to 350 m/d, indicating a considerable variability in the aquifer’s hydraulic parameters. The areas with high hydraulic conductivity were concentrated in the central and eastern parts of the aquifer, and these same areas exhibited a high standard deviation. Based on the findings of this study, while the NSMC method is effective for uncertainty analysis in solving inverse problems, it is important to note that a considerable number of runs are necessary to reach the threshold of calibration error. This is because of the significant non-linearity inherent in the karst aquifer.

Contamination exposure assessment of a well field through Null-Space Monte Carlo method

Contamination exposure assessment of a well field through Null-Space Monte Carlo method

Efficient uncertainty quantification for seawater intrusion prediction using Optimized sampling and Null Space Monte Carlo method

Uncertainty in environmental modeling predictions, stemming from parameter estimation, is a crucial challenge that must be addressed to ensure effective decision-making. Limited field measurements, high computational costs, and a lack of guidance in estimating measurement uncertainty further compound this challenge, particularly for highly parameterized complex models. In this study, we propose a novel and computationally efficient framework for quantifying predictive uncertainty that can be applied to a range of environmental modeling contexts. The novel components of the framework include efficient parameter space sampling using an Optimized Latin hypercube sampling strategy, and applying the Null Space Monte Carlo method (NSMC) along with a developed filtering technique. The NSMC generates sample sets to calibrate the model while exploring the null space. This space contains parameter combinations that are not sufficiently supported by observations. The filtering technique omits low-potential parameter sets from undergoing model calibration. The framework was tested on the seawater intrusion (SWI) model of Wadi Ham aquifer in the United Arab Emirates (UAE) to investigate aquifer sustainability in 2050. Our results demonstrate the importance of incorporating direct and indirect measurements of heads, salinity, and geophysical survey data into the calibration dataset to reduce uncertainty in salinity predictions. The extent of SWI for multiple calibrated parameter sets varied by 4.5% to 11% relative to their means at two main pumping fields. We conclude, with a moderate to a high degree of certainty, that SWI is a serious threat to these fields, and actions are needed to protect the aquifer from salinization. Additionally, variations in SWI length under different geological conditions illustrate regions of high uncertainty that require further data collection. Our framework effectively reduced and quantified prediction uncertainty and provides decision-makers with critical information to inform risk management strategies.

Land subsidence modelling for decision making on groundwater abstraction under emergency situation

Abstract. This study presents an inversion scheme with uncertainty analysis for a land subsidence modelling by a Monte Carlo filter in order to contribute to the decision-making on the groundwater abstraction. For real time prediction and uncertainty analysis under the limited computational resources and available information in emergency situations, one dimensional vertical land subsidence simulation was adopted for the forward modelling and the null-space Monte Carlo method was applied for the effective resampling. The proposed scheme was tested with the existing land subsidence monitoring data in Tokyo lowland, Japan. The results demonstrated that the prediction uncertainty converges and the prediction accuracy improves as the observed data increased with time. The computational time was also confirmed to be acceptable range for a real time execution with a laptop.

The Traveling Pilot Point method. A novel approach to parameterize the inverse problem for categorical fields

Categorical parameter distributions are common-place in hydrogeological systems consisting of geologic facies/categories with distinct properties, e.g., high-permeability channels embedded in a low-permeability matrix. Parameter estimation is difficult in such systems because the discontinuities in the parameter space hinder the inverse problem. Previous research in this area has been focused on the use of stochastic methods. In this paper, we present a novel approach based on Traveling Pilot points (TRIPS) combined with subspace parameter estimation methods to generate realistic categorical parameter distributions that honor calibration constraints (e.g., - measured water levels). In traditional implementations, aquifer properties (e.g., hydraulic conductivity) are estimated at fixed pilot point locations. In the TRIPS implementation, both the properties associated with the pilot points and their locations are estimated. Tikhonov regularization constraints are incorporated in the parameter estimation process to produce realistic parameter depictions. For a synthetic aquifer system, we solved the categorical inverse problem by combining the TRIPS methodology with two subspace methods: Null Space Monte Carlo (NSMC) and Posterior Covariance (PC). A posterior ensemble developed with the rejection sampling (RS) method is compared against the TRIPS ensembles. The comparisons indicated similarities between the various ensembles and to the reference parameter distribution. Between the two subspace methods, the NSMC method produced an ensemble with more variability than the PC method. These preliminary results suggest that the TRIPS methodology has promise and could be tested on more complicated problems.

Evaluation of the performance of a hydraulic barrier by the Null space Monte Carlo method

A gasoline leak caused the contamination of a shallow alluvial aquifer in an urbanized area in Northern Italy. A rapid intervention was conceived to stop the spreading of contamination: a hydraulic barrier has been placed downstream of the source to collect both the floating oil and the contaminated groundwater. A numerical model has been built to assess the performance of the existing barrier, and to design a new configuration of the hydraulic barrier aimed at stopping the hydrocarbon plume already dispersed downstream. A preliminary model was built and calibrated against groundwater levels measured in 41 monitoring wells. Hydraulic conductivities in pilot points, recharge zones and constant head BCs were calibrated. The non-uniqueness of the calibrated parameters led to identify 283 alternative parameter sets, all able to represent the observed heads within an absolute average error of 10 cm. These sets, generated with the Null space Monte Carlo method, served to build 283 models, used to simulate the dispersion of solved contamination through forward particle tracking. A further step was the censoring of all simulations resulting in particle paths at a distance closer than 5 meters from monitoring wells where contamination was never found since the spilled occurred. Analysis was performed of the particle paths generated with the 187 models that were retained. Overall, the barrier captures 89% of all particles. Moreover, in 74% of all realizations, at least a particle escapes, with a mean and median of 7 particles in each realization where it happens. Two main contamination paths are identified: while one is confirmed by the monitoring wells already present, another one would require the placement of new wells to assess the actual presence of contamination. Thus, the validity of the stochastic simulation would be assessed together with the need to improve the performance of the hydraulic barrier.

Probabilistic assessment of the impact of coal seam gas development on groundwater: Surat Basin, Australia

Modelling cumulative impacts of basin-scale coal seam gas (CSG) extraction is challenging due to the long time frames and spatial extent over which impacts occur combined with the need to consider local-scale processes. The computational burden of such models limits the ability to undertake calibration and sensitivity and uncertainty analyses. A framework is presented that integrates recently developed methods and tools to address the computational burdens of an assessment of drawdown impacts associated with rapid CSG development in the Surat Basin, Australia. The null space Monte Carlo method combined with singular value decomposition (SVD)-assisted regularisation was used to analyse the uncertainty of simulated drawdown impacts. The study also describes how the computational burden of assessing local-scale impacts was mitigated by adopting a novel combination of a nested modelling framework which incorporated a model emulator of drawdown in dual-phase flow conditions, and a methodology for representing local faulting. This combination provides a mechanism to support more reliable estimates of regional CSG-related drawdown predictions. The study indicates that uncertainties associated with boundary conditions are reduced significantly when expressing differences between scenarios. The results are analysed and distilled to enable the easy identification of areas where the simulated maximum drawdown impacts could exceed trigger points associated with legislative ‘make good’ requirements; trigger points require that either an adjustment in the development scheme or other measures are implemented to remediate the impact. This report contributes to the currently small body of work that describes modelling and uncertainty analyses of CSG extraction impacts on groundwater.

Null-space Monte Carlo particle tracking to assess groundwater PCE (Tetrachloroethene) diffuse pollution in north-eastern Milan functional urban area

The Lombardy Region in Italy is one of the most urbanized and industrialized areas in Europe. The presence of countless sources of groundwater pollution is therefore a matter of environmental concern. The sources of groundwater contamination can be classified into two different categories: 1) Point Sources (PS), which correspond to areas releasing plumes of high concentrations (i.e. hot-spots) and 2) Multiple-Point Sources (MPS) consisting in a series of unidentifiable small sources clustered within large areas, generating an anthropogenic diffuse contamination. The latter category frequently predominates in European Functional Urban Areas (FUA) and cannot be managed through standard remediation techniques, mainly because detecting the many different source areas releasing small contaminant mass in groundwater is unfeasible. A specific legislative action has been recently enacted at Regional level (DGR IX/3510-2012), in order to identify areas prone to anthropogenic diffuse pollution and their level of contamination. With a view to defining a management plan, it is necessary to find where MPS are most likely positioned. This paper describes a methodology devised to identify the areas with the highest likelihood to host potential MPS. A groundwater flow model was implemented for a pilot area located in the Milan FUA and through the PEST code, a Null-Space Monte Carlo method was applied in order to generate a suite of several hundred hydraulic conductivity field realizations, each maintaining the model in a calibrated state and each consistent with the modelers' expert-knowledge. Thereafter, the MODPATH code was applied to generate back-traced advective flowpaths for each of the models built using the conductivity field realizations. Maps were then created displaying the number of backtracked particles that crossed each model cell in each stochastic calibrated model. The result is considered to be representative of the FUAs areas with the highest likelihood to host MPS responsible for diffuse contamination.

Effect of soil property uncertainties on permafrost thaw projections: a calibration-constrained analysis

Abstract. The effects of soil property uncertainties on permafrost thaw projections are studied using a three-phase subsurface thermal hydrology model and calibration-constrained uncertainty analysis. The null-space Monte Carlo method is used to identify soil hydrothermal parameter combinations that are consistent with borehole temperature measurements at the study site, the Barrow Environmental Observatory. Each parameter combination is then used in a forward projection of permafrost conditions for the 21st century (from calendar year 2006 to 2100) using atmospheric forcings from the Community Earth System Model (CESM) in the Representative Concentration Pathway (RCP) 8.5 greenhouse gas concentration trajectory. A 100-year projection allows for the evaluation of predictive uncertainty (due to soil property (parametric) uncertainty) and the inter-annual climate variability due to year to year differences in CESM climate forcings. After calibrating to measured borehole temperature data at this well-characterized site, soil property uncertainties are still significant and result in significant predictive uncertainties in projected active layer thickness and annual thaw depth-duration even with a specified future climate. Inter-annual climate variability in projected soil moisture content and Stefan number are small. A volume- and time-integrated Stefan number decreases significantly, indicating a shift in subsurface energy utilization in the future climate (latent heat of phase change becomes more important than heat conduction). Out of 10 soil parameters, ALT, annual thaw depth-duration, and Stefan number are highly dependent on mineral soil porosity, while annual mean liquid saturation of the active layer is highly dependent on the mineral soil residual saturation and moderately dependent on peat residual saturation. By comparing the ensemble statistics to the spread of projected permafrost metrics using different climate models, we quantify the relative magnitude of soil property uncertainty to another source of permafrost uncertainty, structural climate model uncertainty. We show that the effect of calibration-constrained uncertainty in soil properties, although significant, is less than that produced by structural climate model uncertainty for this location.

Natural recharge estimation and uncertainty analysis of an adjudicated groundwater basin using a regional-scale flow and subsidence model (Antelope Valley, California, USA)

Groundwater has provided 50–90 % of the total water supply in Antelope Valley, California (USA). The associated groundwater-level declines have led the Los Angeles County Superior Court of California to recently rule that the Antelope Valley groundwater basin is in overdraft, i.e., annual pumpage exceeds annual recharge. Natural recharge consists primarily of mountain-front recharge and is an important component of the total groundwater budget in Antelope Valley. Therefore, natural recharge plays a major role in the Court’s decision. The exact quantity and distribution of natural recharge is uncertain, with total estimates from previous studies ranging from 37 to 200 gigaliters per year (GL/year). In order to better understand the uncertainty associated with natural recharge and to provide a tool for groundwater management, a numerical model of groundwater flow and land subsidence was developed. The transient model was calibrated using PEST with water-level and subsidence data; prior information was incorporated through the use of Tikhonov regularization. The calibrated estimate of natural recharge was 36 GL/year, which is appreciably less than the value used by the court (74 GL/year). The effect of parameter uncertainty on the estimation of natural recharge was addressed using the Null-Space Monte Carlo method. A Pareto trade-off method was also used to portray the reasonableness of larger natural recharge rates. The reasonableness of the 74 GL/year value and the effect of uncertain pumpage rates were also evaluated. The uncertainty analyses indicate that the total natural recharge likely ranges between 34.5 and 54.3 GL/year.

Predictive uncertainty analysis of a saltwater intrusion model using null‐space Monte Carlo

Because of the extensive computational burden and perhaps a lack of awareness of existing methods, rigorous uncertainty analyses are rarely conducted for variable‐density flow and transport models. For this reason, a recently developed null‐space Monte Carlo (NSMC) method for quantifying prediction uncertainty was tested for a synthetic saltwater intrusion model patterned after the Henry problem. Saltwater intrusion caused by a reduction in fresh groundwater discharge was simulated for 1000 randomly generated hydraulic conductivity distributions, representing a mildly heterogeneous aquifer. From these 1000 simulations, the hydraulic conductivity distribution giving rise to the most extreme case of saltwater intrusion was selected and was assumed to represent the “true” system. Head and salinity values from this true model were then extracted and used as observations for subsequent model calibration. Random noise was added to the observations to approximate realistic field conditions. The NSMC method was used to calculate 1000 calibration‐constrained parameter fields. If the dimensionality of the solution space was set appropriately, the estimated uncertainty range from the NSMC analysis encompassed the truth. Several variants of the method were implemented to investigate their effect on the efficiency of the NSMC method. Reducing the dimensionality of the null‐space for the processing of the random parameter sets did not result in any significant gains in efficiency and compromised the ability of the NSMC method to encompass the true prediction value. The addition of intrapilot point heterogeneity to the NSMC process was also tested. According to a variogram comparison, this provided the same scale of heterogeneity that was used to generate the truth. However, incorporation of intrapilot point variability did not make a noticeable difference to the uncertainty of the prediction. With this higher level of heterogeneity, however, the computational burden of generating calibration‐constrained parameter fields approximately doubled. Predictive uncertainty variance computed through the NSMC method was compared with that computed through linear analysis. The results were in good agreement, with the NSMC method estimate showing a slightly smaller range of prediction uncertainty than was calculated by the linear method.