Uncertainties In Boundary Conditions Research Articles

Inverting Topography for Landscape Evolution Model Process Representation: 1. Conceptualization and Sensitivity Analysis

Despite considerable community effort, there is no general set of equations to model long‐term landscape evolution. In order to determine a suitable set of landscape evolution process laws for a site where postglacial erosion has incised valleys up to 50 m deep, we generate a set of alternative models and perform a multimodel analysis. The most basic model we consider includes stream power channel incision, uniform lithology, hillslope transport by linear diffusion, and surface‐water discharge proportional to drainage area. We systematically add one, two, or three elements of complexity to this model from one of four categories: hillslope processes, channel processes, surface hydrology, and representation of geologic materials. We apply methods of formal model analysis to the 37 alternative models. The global Method of Morris sensitivity analysis method is used to identify model input parameters that most and least strongly influence model outputs. Only a few parameters are identified as important, and this finding is consistent across two alternative model outputs: one based on a collection of topographic metrics and one that uses an objective function based on a topographic difference. Parameters that control channel erosion are consistently important, while hillslope diffusivity is important for only select model outputs. Uncertainty in initial and boundary conditions is associated with low sensitivity. Sensitivity analysis provides insight to model dynamics and is a critical step in using model analysis for mechanistic hypothesis testing in landscape evolution theory.

A gaussian process-based iterative Ensemble Kalman Filter for parameter estimation of unsaturated flow

Water flow in the unsaturated zone is an important component of the water cycle. Accurate estimation of soil hydraulic parameters ensures precise simulations of water flow in the unsaturated zone. In this study, a Gaussian Process-based Iterative Ensemble Kalman Filter method (GPIEnKF) is proposed and applied for estimating soil hydraulic parameters for two-dimensional soil water flow. The method involves a surrogate model for two-dimensional soil water flow that is based on the Gaussian Process method. The accuracy and efficiency of the GPIEnKF method are validated using two synthetic cases and a real dataset involving a field experiment with drip irrigation. The impact of the layout of observation points as well as the number of training base points and observation points on the estimation of parameters are also analyzed. The results show that the surrogate model can accurately predict pressure heads at observation points. The layout of observation points that precisely describes infiltration water movement allows the surrogate model to better emulate the original model, thereby improving the accuracy of parameter estimation. The number of training base points is found to have only a small influence on parameter estimation. The accuracy of parameter estimation and pressure head predictions can be further improved by increasing the number of observation points. However, the accuracy of predictions is affected by the uncertainty of boundary conditions and the soil spatial heterogeneity. The GPIEnKF method can effectively estimate multiple parameters characterizing water flow in layered soils. As compared with the standard Iterative Ensemble Kalman Filter, the GPIEnKF method can greatly improve computational efficiency while obtaining comparable results. The GPIEnKF method is an efficient tool for parameter estimation of multi-dimensional soil water flow.

Uncertainty analysis of a loss of cooling and loss of coolant accident in a spent fuel pool using TRACE

In this work the fundamentals and application results of an uncertainty analysis of a loss of coolant and loss of cooling accident in a spent fuel pool are presented. The spent fuel pool was modeled with TRACE V5.0 Patch 5 using a VESSEL 3D component in Cartesian coordinates. A more detailed nodalization was necessary as compared with a previous study of the same accident scenario to better develop the uncertainty analysis in the framework of the Best Estimate Plus Uncertainty (BEPU) approach. First, the model was calibrated using Main Yankee spent fuel pool experimental data of temperatures and flows measured in steady state conditions. Using the calibrated model, the analysis of the accident was performed considering uncertainty in boundary conditions and in all the coefficients included in the TRACE uncertainty quantification data module. The figure of merit selected in this work is the time at which the cladding oxidation reaches 0.17 times the cladding thickness before oxidation (CFR 50.46 b2). The uncertainty analysis was performed using Wilks’ method and surrogate models. In particular, the Elastic Net surrogate model has demonstrated its potential not only to predict the evolution of the figure of merit in an appropriate way with lower computational effort than the Wilks’ method but also its capability to identify the most relevant uncertain parameters, such as the fuel and cladding specific heats, rod internal pressure coefficient, burst temperature coefficient, a group of fluid regime heat transfer coefficients, the cladding metal-water reaction rate coefficient and the cladding tolerance for the average cores. In addition, the surrogate model provides more information of the distribution of the evolution of the figure of merit. The estimations of the 5/95 tolerance limits, both with the Wilks’ method and with the surrogate models, are very similar, ranging from 30.432 to 30.461 h.

Influence of the Arctic on the Predictability of Eurasian Winter Extreme Weather Events

The linkage between the Arctic and midlatitudes has received much attention recently due to the rapidly changing climate. Many investigations have been conducted to reveal the relationship between the Arctic and Eurasian extreme events from the perspective of climatological statistics. As a prediction source for extreme events in Eurasia, Arctic conditions are crucial for extreme event predictions. Therefore, it is urgent to explore the Arctic influence on the predictability of Eurasian extreme events due to the large uncertainties in Arctic conditions. Considering the sensitivity and nonlinearity of the atmospheric circulations in midlatitude to Arctic conditions, it is necessary to investigate the Arctic influences on Eurasian extreme weather events in case studies at weather time scales. Previous studies indicate that only perturbations in specific patterns have fast growth. Thus, the conditional nonlinear optimal perturbation approach is recommended for exploring the uncertainties in Arctic initial and boundary conditions and their synergistic effect on Eurasian extreme events. Moreover, the mechanism for extreme event formation may differ in different cases. Therefore, more extreme cases should be investigated to reach robust conclusions.

Bayesian inference for parameters estimation using experimental data

A finite element (FE) model is developed for a curved cable-stayed footbridge located in Terni (Umbria Region, Central Italy) which accounts for uncertainties in geometry, material properties, and boundary conditions as well as limited knowledge on the behavior of connections and other components. Ambient vibration tests (AVTs) are carried out to identify the main dynamic parameters which are used for model updating in the Bayesian framework. Sensitivity analysis is performed to identify the main mechanical parameters affecting natural frequencies and mode shapes to be used as updating parameters. Finally, the posterior probability distributions of the selected updating parameters is estimated and used to assess the accuracy of the FE-based model. The importance of using a proper informative reference data set in the updating framework is assessed using different observations together with the importance of reliable surrogate models able to reduce the computational costs related to the whole framework.

Bayesian Update and Method of Distributions: Application to Leak Detection in Transmission Mains

AbstractWater‐hammer equations are used to describe transient flow in pipe networks. Uncertainty in model parameters, initial and boundary conditions, and location and strength of a possible leak renders deterministic predictions of this system untenable. When deployed in conjunction with pressure measurements, probabilistic solutions of the water‐hammer equations serve as a tool for detecting leaks in pipes. We use the method of distributions to obtain a probability density function (PDF) for pressure head, whose dynamics are described by the stochastic water‐hammer equations. This PDF provides a prior distribution for subsequent Bayesian data assimilation, in which data collected by pressure sensors are combined with this prior to obtain a posterior PDF of the leak location and size. We conduct a series of numerical experiments with uncertain initial velocity and measurement noise to ascertain the robustness and accuracy of the proposed approach. The results show the method's ability to identify the leak location and strength in a water transmission main.

Improved thermal structure simulation and optimized sampling strategy for Lake Erie using a data assimilative model

Lake Erie has experienced substantial environmental issues (e.g., hypoxia, harmful algal blooms) for decades, which are closely related to the lake’s thermal characteristics. While three-dimensional (3D) hydrodynamic models have been widely applied to Lake Erie, challenges remain due to model representation of physical processes, errors and uncertainty in boundary conditions and forcing terms. The Great Lakes region has a relatively dense and long-term observational record, and these observational data have been used for model initialization and verification, but have not been incorporated into 3D model simulations through data assimilation (DA) to create reanalysis products or improve short-term forecasts. In this work, we developed and evaluated DA to improve thermal structure simulation of Lake Erie. Moored instrument data and satellite data are incorporated into a data-assimilative hydrodynamic model for analysis and evaluation. Results show that DA can effectively improve the model performance to create reanalysis fields when the DA formulation is appropriately developed in recognition of the dynamic complexities and anisotropic error covariances of Lake Erie. The data assimilative model also improves forecasting accuracy and restrains forecasting uncertainty to an acceptable level on a timescale of 1–7 days after being unleashed from DA. Lastly, data sampling strategies based on an error correlation map are examined. Results show the method can effectively reduce the sampling effort while still achieving similar model skills with potential for optimal design of an observation network or field sampling strategy.

Downstream influence of mesoscale convective systems. Part 2: Influence on ensemble forecast skill and spread

Ensemble forecasts are run operationally to determine the forecast uncertainty arising from initial condition, model physics and boundary condition uncertainty. However, global configuration ensembles, which use a convection parametrization scheme, may miss uncertainty because of the misrepresentation of intense convection by such schemes. Here, the impacts of the misrepresentation of mesoscale convective systems (MCSs) on downstream ensemble forecast skill and evolution are determined for a case study. MCS perturbations (calculated‐15 from the difference between outputs from convection‐parametrizing and convection‐permitting Met Office model configurations) are added to six members of a global configuration ensemble created by downscaling forecasts from the global version of the Met Office Global and Regional Ensemble Prediction System. For the first 36 hours, differences grow on the convective scale related to the MCSs, leading to systematic deepening of a developing UK cyclone, although there is damping of the perturbations found in root mean square difference calculations between the forecasts with and without the perturbations (particularly in mean sea level pressure). Subsequently, differences grow rapidly to the synoptic scale, and by five days impact the entire Northern Hemisphere. The MCS perturbations can have systematic effects on ensemble forecasts (e.g., a systematic displacement of a downstream cyclone is found), but, for this case, there is no discernible change in forecast skill as measured by the root mean square error of the ensemble means and the effects of the MCS perturbations are smaller than those generated by the initial condition perturbations. The spread of the combined ensemble (the two ensembles with and without the MCS perturbations) is larger than that of the individual ensembles. Thus, perturbing convection‐parametrizing models to include potential vorticity anomalies associated with MCSs represented in convection‐permitting forecasts, or idealized representations of them, produces alternative realizations from those generated by initial condition perturbations and has the potential to be useful operationally.

Parameters Estimation and Prediction of Water Movement and Solute Transport in Layered, Variably Saturated Soils Using the Ensemble Kalman Filter

The parameters of water movement and solute transport models are essential for the accurate simulation of soil moisture and salinity, particularly for layered soils in field conditions. Parameter estimation can be achieved using the inverse modeling method. However, this type of method cannot fully consider the uncertainties of measurements, boundary conditions, and parameters, resulting in inaccurate estimations of parameters and predictions of state variables. The ensemble Kalman filter (EnKF) is well-suited to data assimilation and parameter prediction in Situations with large numbers of variables and uncertainties. Thus, in this study, the EnKF was used to estimate the parameters of water movement and solute transport in layered, variably saturated soils. Our results indicate that when used in conjunction with the HYDRUS-1D software (University of California Riverside, California, CA, USA) the EnKF effectively estimates parameters and predicts state variables for layered, variably saturated soils. The assimilation of factors such as the initial perturbation and ensemble size significantly affected in the simulated results. A proposed ensemble size range of 50–100 was used when applying the EnKF to the highly nonlinear hydrological models of the present study. Although the simulation results for moisture did not exhibit substantial improvement with the assimilation, the simulation of the salinity was significantly improved through the assimilation of the salinity and relative solutetransport parameters. Reducing the uncertainties in measured data can improve the goodness-of-fit in the application of the EnKF method. Sparse field condition observation data also benefited from the accurate measurement of state variables in the case of EnKF assimilation. However, the application of the EnKF algorithm for layered, variably saturated soils with hydrological models requires further study, because it is a challenging and highly nonlinear problem.

Direct simulation Monte Carlo study of metal evaporation with collimator in e-beam physical vapor deposition

The flow properties and substrate deposition rate profile, which are the important parameters in electron beam physical vapor deposition, are investigated computationally in this article. Collimators are used to achieve the desired vapor beam and deposition rate profile in some applications. This increases the difficulty measuring boundary conditions and the size of the liquid metal pool inside the collimators. It is accordingly hard to obtain accurate results from numerical calculations. In this article, two-dimensional direct simulation Monte Carlo (DSMC) codes are executed to quantify the influence of uncertainties of boundary conditions and pool sizes. Then, three-dimensional DSMC simulations are established to simulate cerium and neodymium evaporation with the collimator. Experimental and computational results of substrate deposition rate profile are in excellent agreement at various evaporation rates and substrate heights. The results show that the DSMC method can assist in metal evaporation with a collimator.

Reanalysis of Ring Current Electron Phase Space Densities Using Van Allen Probe Observations, Convection Model, and Log‐Normal Kalman Filter

AbstractModels of ring current electron dynamics unavoidably contain uncertainties in boundary conditions, electric and magnetic fields, electron scattering rates, and plasmapause location. Model errors can accumulate with time and result in significant deviations of model predictions from observations. Data assimilation offers useful tools which can combine physics‐based models and measurements to improve model predictions. In this study, we systematically analyze performance of the Kalman filter applied to a log‐transformed convection model of ring current electrons and Van Allen Probe data. We consider long‐term dynamics of μ = 2.3 MeV/G and K = 0.3 G1/2RE electrons from 1 February 2013 to 16 June 2013. By using synthetic data, we show that the Kalman filter is capable of correcting errors in model predictions associated with uncertainties in electron lifetimes, boundary conditions, and convection electric fields. We demonstrate that reanalysis retains features which cannot be fully reproduced by the convection model such as storm‐time earthward propagation of the electrons down to 2.5 RE. The Kalman filter can adjust model predictions to satellite measurements even in regions where data are not available. We show that the Kalman filter can adjust model predictions in accordance with observations for μ = 0.1, 2.3, and 9.9 MeV/G and constant K = 0.3 G1/2RE electrons. The results of this study demonstrate that data assimilation can improve performance of ring current models, better quantify model uncertainties, and help deeper understand the physics of the ring current particles.

Structural damage identification incorporating uncertain boundary flexibility by ensemble couple sparse coding classification method

Transfer function (TF) represents the ratio of structure’s response to the loading and can be utilised for the identification of damage in a structure. The capability of distinguishing between changes of the TF due to damage in the existing structure and due to idealised boundary conditions of the analytical model (such as soil-foundation effects or spring bearings) is noteworthy. However, in many cases, having precise knowledge of boundary characteristics is infeasible. In this regard, a number of couple sparse coding (CSC) learnings are utilised with TFs of the structure with various probable boundary conditions as input parameters to reduce the uncertainty of idealised boundary conditions. The system of CSCs termed ensemble to enhance efficiency of the method. Principle component analysis (PCA) technique is applied to avoid over fitting problem due to the large size of TF data in machine learning training. In numerical verifications, the damage scenarios are considered different from the trained data. A 2 D frame structure is presented as numerical example to demonstrate the adequacy of the proposed method for damage detection under complicated effects of soil foundation. A beam which is suspended using soft rubber bands with uncertain stiffness is considered for experimental validation of the method. Results indicate that the proposed method is capable of identifying damage properties with acceptable precision.

Uncertainty quantification of computational coronary stenosis assessment and model based mitigation of image resolution limitations

Coronary artery disease is one of the leading causes of death globally. The hallmark of this disease is the occurrence of stenosed coronary arteries which reduce blood flow to the myocardium. Severely stenosed arteries can be treated if detected, but the diagnostic procedure to assess fractional flow reserve (FFR), a quantitative measure of stenosis severity, is invasive, burdensome to the patient, and costly. Recent computational approaches estimate the severity of stenoses from simulations of coronary blood flow based on CT imagery. These methods allow for diagnosis to be made noninvasively and using fewer hospital resources; however, the predictions depend on uncertain input data and model parameters due to technical limitations and patient variability. To assess the consequences of boundary condition and input uncertainty on predictions of FFR, we developed a model of coronary blood flow. We performed uncertainty quantification and sensitivity analysis of the predictions based on uncertainties in boundary conditions, parameters, and geometric measurements. Our results identified three influential sources of uncertainty: geometric data, cardiac output, and coronary resistance during hyperemia. Further, uncertainty about the geometry of the stenosed coronary branch influences estimates much more than other geometrical data. Limitations of medical imaging contribute uncertainty to predictions as vessels below a certain threshold remain unobserved. We assessed the effects of unobserved vessels by comparing predictions based on both high and low resolution data. Moreover, we introduced a novel method that estimates flow distribution while accounting for unobserved vessels. This method improved FFR predictions in the cases considered by 50% on average.

Exploration of Antarctic Ice Sheet 100-year contribution to sea level rise and associated model uncertainties using the ISSM framework

Abstract. Estimating the future evolution of the Antarctic Ice Sheet (AIS) is critical for improving future sea level rise (SLR) projections. Numerical ice sheet models are invaluable tools for bounding Antarctic vulnerability; yet, few continental-scale projections of century-scale AIS SLR contribution exist, and those that do vary by up to an order of magnitude. This is partly because model projections of future sea level are inherently uncertain and depend largely on the model's boundary conditions and climate forcing, which themselves are unknown due to the uncertainty in the projections of future anthropogenic emissions and subsequent climate response. Here, we aim to improve the understanding of how uncertainties in model forcing and boundary conditions affect ice sheet model simulations. With use of sampling techniques embedded within the Ice Sheet System Model (ISSM) framework, we assess how uncertainties in snow accumulation, ocean-induced melting, ice viscosity, basal friction, bedrock elevation, and the presence of ice shelves impact continental-scale 100-year model simulations of AIS future sea level contribution. Overall, we find that AIS sea level contribution is strongly affected by grounding line retreat, which is driven by the magnitude of ice shelf basal melt rates and by variations in bedrock topography. In addition, we find that over 1.2 m of AIS global mean sea level contribution over the next century is achievable, but not likely, as it is tenable only in response to unrealistically large melt rates and continental ice shelf collapse. Regionally, we find that under our most extreme 100-year warming experiment generalized for the entire ice sheet, the Amundsen Sea sector is the most significant source of model uncertainty (1032 mm 6σ spread) and the region with the largest potential for future sea level contribution (297 mm). In contrast, under a more plausible forcing informed regionally by literature and model sensitivity studies, the Ronne basin has a greater potential for local increases in ice shelf basal melt rates. As a result, under this more likely realization, where warm waters reach the continental shelf under the Ronne ice shelf, it is the Ronne basin, particularly the Evans and Rutford ice streams, that are the greatest contributors to potential SLR (161 mm) and to simulation uncertainty (420 mm 6σ spread).

Comparison of data-driven uncertainty quantification methods for a carbon dioxide storage benchmark scenario

A variety of methods is available to quantify uncertainties arising with\-in the modeling of flow and transport in carbon dioxide storage, but there is a lack of thorough comparisons. Usually, raw data from such storage sites can hardly be described by theoretical statistical distributions since only very limited data is available. Hence, exact information on distribution shapes for all uncertain parameters is very rare in realistic applications. We discuss and compare four different methods tested for data-driven uncertainty quantification based on a benchmark scenario of carbon dioxide storage. In the benchmark, for which we provide data and code, carbon dioxide is injected into a saline aquifer modeled by the nonlinear capillarity-free fractional flow formulation for two incompressible fluid phases, namely carbon dioxide and brine. To cover different aspects of uncertainty quantification, we incorporate various sources of uncertainty such as uncertainty of boundary conditions, of conceptual model definitions and of material properties. We consider recent versions of the following non-intrusive and intrusive uncertainty quantification methods: arbitary polynomial chaos, spatially adaptive sparse grids, kernel-based greedy interpolation and hybrid stochastic Galerkin. The performance of each approach is demonstrated assessing expectation value and standard deviation of the carbon dioxide saturation against a reference statistic based on Monte Carlo sampling. We compare the convergence of all methods reporting on accuracy with respect to the number of model runs and resolution. Finally we offer suggestions about the methods' advantages and disadvantages that can guide the modeler for uncertainty quantification in carbon dioxide storage and beyond.

An Innovative Approach to Minimizing Uncertainty in Sediment Load Boundary Conditions for Modelling Sedimentation in Reservoirs

A number of significant investigations have advanced our understanding of the parameters influencing reservoir sedimentation. However, a reliable modelling of sediment deposits and delta formation in reservoirs is still a challenging problem due to many uncertainties in the modelling process. Modelling performance can be improved by adjusting the uncertainty caused by sediment load boundary conditions. In our study, we diminished the uncertainty factor by setting more precise sediment load boundary conditions reconstructed using wavelet artificial neural networks for a morphodynamic model. The model was calibrated for hydrodynamics using a backward error propagation method. The proposed approach was applied to the Tarbela Reservoir located on the Indus River, in northern Pakistan. The results showed that the hydrodynamic calibration with coefficient of determination (R2) = 0.969 and Nash–Sutcliffe Efficiency (NSE) = 0.966 also facilitated good calibration in morphodynamic calculations with R2 = 0.97 and NSE = 0.96. The model was validated for the sediment deposits in the reservoir with R2 = 0.96 and NSE = 0.95. Due to desynchronization between the glacier melts and monsoon rain caused by warmer climate and subsequent decrease of 17% in sediment supply to the Tarbela dam, our modelling results showed a slight decrease in the sediment delta for the near future (until 2030). Based on the results, we conclude that our overall state-of-the-art modelling offers a significant improvement in computational time and accuracy, and could be used to estimate hydrodynamic and morphodynamic parameters more precisely for different events and poorly gauged rivers elsewhere in the world. The modelling concept could also be used for predicting sedimentation in the reservoirs under sediment load variability scenarios.

Probabilistic forecasting of wind power production losses in cold climates: a case study

Abstract. The problem of icing on wind turbines in cold climates is addressed using probabilistic forecasting to improve next-day forecasts of icing and related production losses. A case study of probabilistic forecasts was generated for a 2-week period. Uncertainties in initial and boundary conditions are represented with an ensemble forecasting system, while uncertainties in the spatial representation are included with a neighbourhood method. Using probabilistic forecasting instead of one single forecast was shown to improve the forecast skill of the ice-related production loss forecasts and hence the icing forecasts. The spread of the multiple forecasts can be used as an estimate of the forecast uncertainty and of the likelihood for icing and severe production losses. Best results, both in terms of forecast skill and forecasted uncertainty, were achieved using both the ensemble forecast and the neighbourhood method combined. This demonstrates that the application of probabilistic forecasting for wind power in cold climates can be valuable when planning next-day energy production, in the usage of de-icing systems and for site safety.

Stochastic modelling of hygrothermal performance of highly insulated wood framed walls

As energy consumption has become an important issue in building design, most building codes require a higher insulation level for building envelopes to improve the building's energy efficiency. However, the highly insulated walls may lead to a higher risk of moisture problems. Although hygrothermal simulation has been widely used to investigate the moisture performance of wood framed walls, the uncertainties of input parameters such as material properties, boundary conditions and moisture loads, may lead to discrepancies between simulation results and actual performance of the envelope. This paper investigates the hygrothermal performance of highly insulated wood framed walls using a stochastic approach, which combines the Latin Hypercube Sampling method and Factorial Design to take into account the uncertainties of material properties, boundary conditions and moisture loads (air leakage and rain leakage). The investigated walls include an I-joist deep cavity wall, two exterior insulated walls, and a conventional 2 × 6 stud wall as the baseline. It is found that under the moisture loads introduced (i.e. air leakage and rain leakage), the exterior insulated walls have a lower mold growth risk than the deep cavity wall and the 2 × 6 stud baseline wall. The uncertainties of material properties do not result in significant variations in simulation results such as moisture content and mold growth index as uncertainties of moisture loads do. The hygrothermal performance of these highly insulated walls is more sensitive to moisture loads and the significance of the moisture loads (air leakage and rain leakage) depends on climatic conditions.

A probabilistic framework for floodplain mapping using hydrological modeling and unsteady hydraulic modeling

ABSTRACTPrediction of design hydrographs is key in floodplain mapping using hydraulic models, which are either steady state or unsteady. The former, which require only an input peak, substantially overestimate the volume of water entering the floodplain compared to the more realistic dynamic case simulated by the unsteady models that require the full hydrograph. Past efforts to account for the uncertainty of boundary conditions using unsteady hydraulic modeling have been based largely on a joint flood frequency–shape analysis, with only a very limited number of studies using hydrological modeling to produce the design hydrographs. This study therefore presents a generic probabilistic framework that couples a hydrological model with an unsteady hydraulic model to estimate the uncertainty of flood characteristics. The framework is demonstrated on the Swannanoa River watershed in North Carolina, USA. Given its flexibility, the framework can be applied to study other sources of uncertainty in other hydrological models and watersheds.

Constraints from zoom-in simulations on the protostellar accretion process

AbstractStars are embedded in different environments of Giant Molecular Clouds during their formation phase. Despite this fact, it is common practice to assume an isolated spherical core as the initial condition for models of individual star formation. To avoid the uncertainties of initial and boundary conditions, we use an alternative approach of zoom-in simulations to account for the environment in which protostars form. Our models show that injections of 26Al from a close-by supernova into the young solar system were highly unlikely. Moreover, we find that the accretion process of protostars is heterogeneous and environment-dependent.