Bayesian Calibration and Sensitivity Analysis for a Karst Aquifer Model Using Active Subspaces

In this article, we perform a parameter study for a recently developed karst hydrological model. The study consists of a high-dimensional Bayesian inverse problem and a global sensitivity analysis. For the first time in karst hydrology, we use the active subspace method to find directions in the parameter space that dominate the Bayesian update from the prior to the posterior distribution in order to effectively reduce the dimension of the problem and for computational efficiency. Additionally, the calculated active subspace can be exploited to construct sensitivity metrics on each of the individual parameters and be used to construct a natural model surrogate. The model consists of 21 parameters to reproduce the hydrological behavior of spring discharge in a karst aquifer located in the Kerschbaum spring recharge area at Waidhofen a.d. Ybbs in Austria. The experimental spatial and time series data for the inference process were collected by the water works in Waidhofen. We show that this case study has implicit low-dimensionality, and we run an adjusted Markov chain Monte Carlo algorithm in a low-dimensional subspace to construct samples of the posterior distribution. The results are visualized and verified by plots of the posterior's push-forward distribution displaying the uncertainty in predicting discharge values due to the experimental noise in the data. Finally, a discussion provides hydrological interpretation of these results for the Kerschbaum area.

[1]  Andrew Gelman,et al.  Handbook of Markov Chain Monte Carlo , 2011 .

[2]  Damir Jukić,et al.  Groundwater balance estimation in karst by using a conceptual rainfall-runoff model , 2009 .

[3]  Thorsten Wagener,et al.  Process-based karst modelling to relate hydrodynamic and hydrochemical characteristics to system properties , 2013 .

[4]  Soroosh Sorooshian,et al.  Evaluating model performance and parameter behavior for varying levels of land surface model complexity , 2006 .

[5]  J. Norris.  Appendix: probability and measure , 1997 .

[6]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[7]  I. Sobola,et al.  Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates , 2001 .

[8]  R. Ghanem,et al.  Uncertainty propagation using Wiener-Haar expansions , 2004 .

[9]  Thorsten Wagener,et al.  Testing the realism of model structures to identify karst system processes using water quality and quantity signatures , 2013 .

[10]  Karen Willcox,et al.  Multifidelity Dimension Reduction via Active Subspaces , 2018, SIAM J. Sci. Comput..

[11]  R. Ghanem,et al.  Multi-resolution analysis of wiener-type uncertainty propagation schemes , 2004 .

[12]  Heikki Haario,et al.  DRAM: Efficient adaptive MCMC , 2006, Stat. Comput..

[13]  Trent Michael Russi,et al.  Uncertainty Quantification with Experimental Data and Complex System Models , 2010 .

[14]  Rachid Ababou,et al.  Linear and nonlinear input/output models for karstic springflow and flood prediction at different time scales , 1999 .

[15]  Sabine Attinger,et al.  The impact of standard and hard‐coded parameters on the hydrologic fluxes in the Noah‐MP land surface model , 2016 .

[16]  Jasper A. Vrugt,et al.  Markov chain Monte Carlo simulation using the DREAM software package: Theory, concepts, and MATLAB implementation , 2016, Environ. Model. Softw..

[17]  George Kuczera,et al.  Bayesian analysis of input uncertainty in hydrological modeling: 1. Theory , 2006 .

[18]  Paul G. Constantine,et al.  Active subspaces for sensitivity analysis and dimension reduction of an integrated hydrologic model , 2015, Comput. Geosci..

[19]  J. Tinsley Oden,et al.  Solution verification, goal-oriented adaptive methods for stochastic advection-diffusion problems , 2010 .

[20]  A. Jakeman,et al.  How much complexity is warranted in a rainfall‐runoff model? , 1993 .

[21]  J. Nash,et al.  River flow forecasting through conceptual models part I — A discussion of principles☆ , 1970 .

[22]  I. Padilla,et al.  Characterization, modeling, and remediation of karst in a changing environment , 2018, Environmental Earth Sciences.

[23]  Vincent Guinot,et al.  KarstMod: A modelling platform for rainfall - discharge analysis and modelling dedicated to karst systems , 2017, Environ. Model. Softw..

[24]  T. Reimann,et al.  MODFLOW‐CFP: A New Conduit Flow Process for MODFLOW–2005 , 2009 .

[25]  Andrew M. Stuart,et al.  Inverse problems: A Bayesian perspective , 2010, Acta Numerica.

[26]  Ilias Bilionis,et al.  Gaussian processes with built-in dimensionality reduction: Applications in high-dimensional uncertainty propagation , 2016, 1602.04550.

[27]  P. Kitanidis,et al.  Principal Component Geostatistical Approach for large-dimensional inverse problems , 2014, Water resources research.

[28]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[29]  J. Vrugt,et al.  Inverse Modeling of Subsurface Flow and Transport Properties: A Review with New Developments , 2008 .

[30]  M. Sauter,et al.  Modellierung der Hydraulik von Karstgrundwasserleitern – Eine Übersicht , 2006 .

[31]  William A. Link,et al.  On thinning of chains in MCMC , 2012 .

[32]  Barbara Wohlmuth,et al.  Goal-oriented adaptive surrogate construction for stochastic inversion , 2018, Computer Methods in Applied Mechanics and Engineering.

[33]  Zong-Liang Yang,et al.  Quantifying parameter sensitivity, interaction, and transferability in hydrologically enhanced versions of the Noah land surface model over transition zones during the warm season , 2010 .

[34]  M. Disse,et al.  Modeling the hydrological impact of land use change in a dolomite-dominated karst system , 2018, Journal of Hydrology.

[35]  F. Pianosi,et al.  V2Karst V1.1: a parsimonious large-scale integrated vegetation–recharge model to simulate the impact of climate and land cover change in karst regions , 2018, Geoscientific Model Development.

[36]  Damir Jukić,et al.  Estimating parameters of groundwater recharge model in frequency domain: Karst springs Jadro and Žrnovnica , 2008 .

[37]  M. Girolami,et al.  Solving large-scale PDE-constrained Bayesian inverse problems with Riemann manifold Hamiltonian Monte Carlo , 2014, 1407.1517.

[38]  Thorsten Wagener,et al.  Karst water resources in a changing world: Review of hydrological modeling approaches , 2014 .

[39]  Johan Larsson,et al.  Exploiting active subspaces to quantify uncertainty in the numerical simulation of the HyShot II scramjet , 2014, J. Comput. Phys..

[40]  L. Gottschalk,et al.  Bayesian estimation of parameters in a regional hydrological model , 2002 .

[41]  Keith Beven,et al.  A manifesto for the equifinality thesis , 2006 .

[42]  D. Labat,et al.  Dynamics of the Flow Exchanges between Matrix and Conduits in Karstified Watersheds at Multiple Temporal Scales , 2019, Water.

[43]  Georg Teutsch,et al.  Modelling karst groundwater hydraulics An overview , 2006 .

[44]  P. Martínez-Santos,et al.  Simplified VarKarst Semi-distributed Model Applied to Joint Simulations of Discharge and Piezometric Variations in Villanueva Del Rosario Karst System (Malaga, Southern Spain) , 2019, Advances in Karst Science.

[45]  George E. Karniadakis,et al.  Beyond Wiener–Askey Expansions: Handling Arbitrary PDFs , 2006, J. Sci. Comput..

[46]  Joel A. Tropp,et al.  User-Friendly Tail Bounds for Sums of Random Matrices , 2010, Found. Comput. Math..

[47]  Patrick Willems,et al.  Global sensitivity analysis of yield output from the water productivity model , 2014, Environ. Model. Softw..

[48]  Shubhangi Gupta,et al.  Efficient parameter estimation for a methane hydrate model with active subspaces , 2018, Computational Geosciences.

[49]  Andreas Hartmann,et al.  On the value of water quality data and informative flow states in karst modelling , 2017 .

[50]  D. Gleich,et al.  Computing active subspaces with Monte Carlo , 2014, 1408.0545.

[51]  R. Plessix.  A review of the adjoint-state method for computing the gradient of a functional with geophysical applications , 2006 .

[52]  Qiqi Wang,et al.  Erratum: Active Subspace Methods in Theory and Practice: Applications to Kriging Surfaces , 2013, SIAM J. Sci. Comput..

[53]  Paul G. Constantine,et al.  Accelerating Markov Chain Monte Carlo with Active Subspaces , 2016, SIAM J. Sci. Comput..

[54]  Michael Bruen,et al.  Understanding hydrological flow paths in conceptual catchment models using uncertainty and sensitivity analysis , 2016, Comput. Geosci..

[55]  D. Higdon,et al.  Accelerating Markov Chain Monte Carlo Simulation by Differential Evolution with Self-Adaptive Randomized Subspace Sampling , 2009 .

[56]  Fabio Nobile,et al.  A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2008, SIAM J. Numer. Anal..

[57]  C. W. Thornthwaite.  An approach toward a rational classification of climate. , 1948 .

[58]  W. Graham,et al.  What Makes a First‐Magnitude Spring?: Global Sensitivity Analysis of a Speleogenesis Model to Gain Insight into Karst Network and Spring Genesis , 2018, Water Resources Research.

[59]  Paul G. Constantine,et al.  Global sensitivity metrics from active subspaces , 2015, Reliab. Eng. Syst. Saf..

[60]  Paul G. Constantine,et al.  Time‐dependent global sensitivity analysis with active subspaces for a lithium ion battery model , 2016, Stat. Anal. Data Min..

[61]  H. Jourde,et al.  Modelling the hydrologic functions of a karst aquifer under active water management – The Lez spring , 2009 .

[62]  Tiangang Cui,et al.  Dimension-independent likelihood-informed MCMC , 2014, J. Comput. Phys..

[63]  Paul G. Constantine,et al.  A modified SEIR model for the spread of Ebola in Western Africa and metrics for resource allocation , 2016, Appl. Math. Comput..

[64]  M. Trosset,et al.  Bayesian recursive parameter estimation for hydrologic models , 2001 .

[65]  Cajo J. F. ter Braak,et al.  Treatment of input uncertainty in hydrologic modeling: Doing hydrology backward with Markov chain Monte Carlo simulation , 2008 .

[66]  Hoshin Vijai Gupta,et al.  Decomposition of the mean squared error and NSE performance criteria: Implications for improving hydrological modelling , 2009 .

[67]  W. Yeh,et al.  Parameter-independent model reduction of transient groundwater flow models: Application to inverse problems , 2014 .

[68]  J. Maréchal,et al.  Turbulent and Laminar Flow in Karst Conduits Under Unsteady Flow Conditions: Interpretation of Pumping Tests by Discrete Conduit‐Continuum Modeling , 2018 .

[69]  H. Jourde,et al.  KARSTMOD: A Generic Modular Reservoir Model Dedicated to Spring Discharge Modeling and Hydrodynamic Analysis in Karst , 2014 .

[70]  Ilse C. F. Ipsen,et al.  A Probabilistic Subspace Bound with Application to Active Subspaces , 2018, SIAM J. Matrix Anal. Appl..

[71]  D. Labat,et al.  Rainfall runoff relations for karstic springs. Part I: convolution and spectral analyses , 2000 .